Algebraic Expressions

Transcription

1 5th Year Maths Higher Level Kieran Mills Algebraic Expressions No part of this publication may be copied, reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from The Dublin School of Grinds. Ref: 5/maths/h/lc/Algebraic Expressions notes

2 EASTER REVISION COURSES Looking to maximise your CAO points? Easter is a crucial time for students to vastly improve on the points that they received in their mock exams. To help students take advantage of this valuable time, The Dublin School of Grinds is running intensive, examfocused Easter Revision Courses. Each course runs for five days (90 minutes per day). All courses take place in Stillorgan, Co. Dublin. The focus of these courses is to maximise students CAO points. EASTER REVISION COURSE FEES: 6TH YEAR & 5TH YEAR COURSES PRICE TOTAL SAVINGS 1st Course nd Course rd Course FREE th Course th Course th Course ,05 7th Course ,0 8th Course ,465 9th Course ,710 RD YEAR COURSES PRICE TOTAL SAVINGS SPECIAL OFFER 1st Course nd Course To avail of this offer, early booking is required as courses were fully booked last year. rd Course FREE th Course th Course th Course th Course th Course BUY COURSES GET A RD COURSE FREE What do students get at these courses? minutes of intensive tuition per day for five days, with Ireland s leading teachers. 99 Comprehensive study notes. 99 A focus on simple shortcuts to raise students grades and exploit the critically important marking scheme. 99 Access to a free supervised study room. NOTE: These courses are built on the fact that there are certain predicable trends that reappear over and over again in the State Examinations. DSOG Easter 017 8pg A4 FINAL PRINT.indd 5% SIBLING DISCOUNT AVAILABLE. Please call to avail of this discount. FREE DAILY BUS SERVICE For full information on our Easter bus service, see pages ahead. Access to food and beverage facilities is also available to students. To book, call us on or book online at 0/0/017 1:5

3 Timetable An extensive range of course options are available over a two-week period to cater for students timetable needs. Courses are held over the following weeks:» Monday 10th Friday 14th April 017» Monday 17th Friday 1st April 017 All Easter Revision Courses take place in The Talbot Hotel, Stillorgan (formerly known as The Stillorgan Park Hotel). 6th Year Easter Revision Courses SUBJECT LEVEL DATES TIME Accounting H Monday 10th - Friday 14th April 1:00pm - 1:0pm Agricultural Science H Monday 10th - Friday 14th April 10:00am - 11:0am Applied Maths H Monday 10th - Friday 14th April 8:00am - 9:0am Art History H Monday 10th - Friday 14th April :00pm - :0pm Biology Course A* H Monday 10th - Friday 14th April 8:00am - 9:0am Biology Course A* H Monday 17th - Friday 1st April 10:00am - 11:0am Biology Course B* H Monday 10th - Friday 14th April 10:00am - 11:0am Biology Course B* H Monday 17th - Friday 1st April 8:00am - 9:0am Business H Monday 10th - Friday 14th April 1:00pm - 1:0pm Business H Monday 17th - Friday 1st April 8:00am - 9:0am Chemistry Course A* H Monday 17th - Friday 1st April 8:00am - 9:0am Chemistry Course B* H Monday 17th - Friday 1st April 10:00am - 11:0am Classical Studies H Monday 10th - Friday 14th April :00pm - :0pm Economics H Monday 10th - Friday 14th April 8:00am - 9:0am Economics H Monday 17th - Friday 1st April 10:00am - 11:0am English Paper 1* H Monday 17th - Friday 1st April 8:00am - 9:0am English Paper * H Monday 10th - Friday 14th April 8:00am - 9:0am English Paper * H Monday 17th - Friday 1st April 10:00am - 11:0am French H Monday 10th - Friday 14th April 10:00am - 11:0am French H Monday 17th - Friday 1st April 8:00am - 9:0am Geography H Monday 10th - Friday 14th April 8:00am - 9:0am Geography H Monday 10th - Friday 14th April :00pm - :0pm German H Monday 17th - Friday 1st April 1:00pm - 1:0pm History (Europe)* H Monday 17th - Friday 1st April :00pm - :0pm History (Ireland)* H Monday 17th - Friday 1st April 1:00pm - 1:0pm Home Economics H Monday 10th - Friday 14th April 1:00pm - 1:0pm Irish H Monday 10th - Friday 14th April 10:00am - 11:0am Irish H Monday 17th - Friday 1st April 1:00pm - 1:0pm Maths Paper 1* H Monday 10th - Friday 14th April 8:00am - 9:0am Maths Paper 1* H Monday 10th - Friday 14th April 1:00pm - 1:0pm Maths Paper 1* H Monday 17th - Friday 1st April 8:00am - 9:0am Maths Paper * H Monday 10th - Friday 14th April 10:00am - 11:0am Maths Paper * H Monday 17th - Friday 1st April 10:00am - 11:0am Maths Paper * H Monday 17th - Friday 1st April 1:00pm - 1:0pm Maths O Monday 10th - Friday 14th April 10:00am - 11:0am Maths O Monday 17th - Friday 1st April 1:00pm - 1:0pm Physics H Monday 17th - Friday 1st April 10:00am - 11:0am Spanish H Monday 10th - Friday 14th April 1:00pm - 1:0pm Spanish H Monday 17th - Friday 1st April 10:00am - 11:0am Note: 5th Year students are welcome to attend any of the 6th Year courses above. * Due to large course content, these subjects have been divided into two courses. For a full list of topics covered in these courses, please see pages ahead. To book, call us on or book online at 5th Year Easter Revision Courses SUBJECT LEVEL DATES TIME English H Monday 10th - Friday 14th April 1:00pm - 1:0pm Maths H Monday 10th - Friday 14th April 10:00am - 11:0am Note: 4th Year students are welcome to attend any of the 5th Year courses listed above. rd Year Easter Revision Courses SUBJECT LEVEL DATES TIME Business H Monday 17th - Friday 1st April :00pm - :0pm Studies English H Monday 10th - Friday 14th April 10:00am - 11:0am English H Monday 17th - Friday 1st April 1:00pm - 1:0pm French H Monday 17th - Friday 1st April 1:00pm - 1:0pm Geography H Monday 17th - Friday 1st April 8:00am - 9:0am German H Monday 17th - Friday 1st April :00pm - :0pm History H Monday 10th - Friday 14th April 8:00am - 9:0am Irish H Monday 10th - Friday 14th April 1:00pm - 1:0pm Maths H Monday 10th - Friday 14th April 8:00am - 9:0am Maths H Monday 17th - Friday 1st April 10:00am - 11:0am Maths O Monday 10th - Friday 14th April :00pm - :0pm Science H Monday 10th - Friday 14th April 1:00pm - 1:0pm Science H Monday 17th - Friday 1st April 8:00am - 9:0am Spanish H Monday 10th - Friday 14th April :00pm - :0pm Note: nd Year students are welcome to attend any of the rd Year courses above. 6th Year Oral Preparation Courses With the Oral marking component worth up to 40%, it is of paramount importance that students are fully prepared for these examinations. These courses will show students how to lead the Examiner towards topics they are prepared for. This will equip students with the information they need to maximise their performance in the State Examinations. FEES: 140 VENUE: The Talbot Hotel, Stillorgan (formerly The Stillorgan Park Hotel) SUBJECT LEVEL DATES TIME French H Sunday 1th March 9:00am - 1:00pm German H Saturday 11th March 9:00am - 1:00pm Irish H Sunday 19th March 9:00am - 1:00pm Spanish H Saturday 11th March :00pm - 6:00pm BUY COURSES & GET A RD COURSE FREE!

4 Section : Algebraic Expressions Paper 1 Topics Section 1. Number Section. Algebraic expressions Section. Algebraic equations Section 4. Sequences and Series Section 5. Financial Maths Section 6. Complex Numbers Section 7. Functions Section 8. Differentiation Section 9. Integration Section 10: Proof by Induction Paper Topics Section 1. Geometry Section. Measurement Section. Trigonometry Section 4. Co-ordinate Geometry Section 5. Probability Section 6. Statistics Contents Chapter : Working with Algebraic Expressions... Chapter 4: Polynomial and Rational Expressions...18 Chapter 5: Exponentials and Logs...9 Exercises...9 Answers...58 Leaving Certificate Exam questions (All the algebra questions are at the end of Section ) The Dublin School of Grinds Page 1 Mills & Kelly (Power of Maths)

5 Chapter. Working with Algebraic Expressions 1. Algebraic Basics Some important key terms: x - xy + 5 is an expression. x, xy and 5 are terms. This expression has three terms. x and y are variables. x and y can take on different values. In the term x, is the coefficient of the term. In the term xy, is the coefficient of the term. 5 is a constant. Example 1: Write down the number of terms, the coefficient of a given term and the constant term in the following algebraic expressions: Number of terms Coefficient Constant (a) 4 + x + 4xy (b) x y + 5xy - (c) 4ab - a b + 5a b - 5 (d) x - x + x xy x y a b x. Combining like terms Some important points: 1. The order in which you multiply two numbers does not matter. x y y x xy yx The big and the little : Write out in long hand:. The coefficient in front of the variable tells you how many times you add the variable to itself. x x + x + x 5x x + x + x + x + x The Dublin School of Grinds Page Mills & Kelly (Power of Maths)

6 . Combining powers: The little number (power) tells you how many times you multiply a number by itself. x x x x x x x x x x x x 5 x + Example : Multiply the following terms together to produce a single term (a) 4x 5y (b) 5ab 4ab (c) (-xy)(-5x) (d) x x 4 (e) xy -4x y a p a q a p + q You only add like terms. x + x x + x + x + x + x 5x [ terms are combined into 1 term] a b + 5a b - ba 6a b [ terms are combined into 1 term] But x + y x + y [ terms remain as terms] Example : Simplify the following. (a) a + a + 5a (b) 4x - 5x - 7x (c) 4x + y + y - 10x (d) 4(a - b) - (b - a) (e) x y + 4x y - 11yx You can only add like terms. Unlike terms can be multiplied. 4ab + ab 4ab ab The Dublin School of Grinds Page Mills & Kelly (Power of Maths)

7 . Multiplying out Brackets ( x+ y)( x y) x( x y) + y( x y) 4x xy+ 6yx y 4x xy+ 6xy y 4x + 4xy y (x + y)(x - y) 4x + 4xy - y (x + y) and (x - y) multiply together to give 4x + 4xy - y. (x + y) and (x - y) are called the factors of 4x + 4xy - y. Example 4: Expand the following. (a) (a + b)(c + d) Number of terms: (b) (a + b)(a - b) Number of terms: Trinomial (c) (a + b)(a - b) Number of terms: Difference of squares Special situations 1. Difference of two squares: (Difference of two terms) (Sum of the same two terms) ( x 5y)( x+ 5y) x( x+ 5y) 5y( x+ 5y) 4x + 10xy 10xy 5y 4x 5y ( x) ( 5y) Remember as: (First - Second)(First + Second) (First) - (Second) The Dublin School of Grinds Page 4 Mills & Kelly (Power of Maths)

8 Example 5: Expand the following. (a) (x + y)(x - y) (b) (5x + y)(5x - y) (c) (ax + by)(ax - by). Perfect square: Consists of two identical brackets. ( x y) ( x y)( x y) x( x y) y( x y) 9x 6xy 6xy+ 4y 9x 1xy+ 4y Example 6: Expand the following. (a) (x + y) ( x) + ( x)( y) + ( y) Remember as: (First term + Second term) (First term) + (First term)(second term) + (Second term) (b) (x - 4y) (c) (ax + by) The Dublin School of Grinds Page 5 Mills & Kelly (Power of Maths)

9 . Harder multiplication Example 7: Expand the following. (a) (x - 1)(x + )(x + 1) (b) (x - ) 4. Finding the value of an algebraic expression To find the value of an algebraic expression, simply fill in the given value(s) of the varible(s). Example 8: Evaluate 4x y - xy + 5xy - 7, if x and y -. For you to practice Exercise 1: page 9 The Dublin School of Grinds Page 6 Mills & Kelly (Power of Maths)

10 . Binomial Theorem An expression with two terms is called a binomial. The binomial theorem is a quick way to multiply out (expand) binomials raised to a power. Examples: (x + y) 6, (a - b) How n C r is calculated Factorials Factorials are represented by! Example: 4! Doing factorials on the calculator: Casio fx-8gt PLUS Example: 4! Press 4 Press SHIFT x -1 (This is x!) Press Calculate the following manually and then check the answers on your calculator: 5! 6! 1! Write out: n! Combinations We will deal with combinations (C) or selections in greater detail when we do the section on Probability. n C r n! r!( n-r)! Formula and Tables Book: Page 0 (Algebra) Example: How many way can you select two people from 10 people when forming a committee? n C r gives you the answer where n 10, r ! C 10-! 8!( )!!! C C 6 9 The Dublin School of Grinds Page 7 Mills & Kelly (Power of Maths)

11 10 C 4 1 C Alternative way of writing combinations: n C Doing combinations on the calculator: Casio fx-8gt PLUS Example: 1 C Press 1 Press SHIFT (This is ncr) Press Press r n r. Using the Binomial Formula and Tables Book: Page 0 (Algebra) n n n ( x y) x n n x n + + y + -1 x 0 1 Example 9: Expand (x + y) 5. y n r x y n n y n- n-r r n Handy coefficients Coefficients of (x + y) : 1,, 1 Coefficients of (x + y) : 1,,, 1 Coefficients of (x + y) 4 : 1, 4, 6, 4, 1 Coefficients of (x + y) 5 : The Dublin School of Grinds Page 8 Mills & Kelly (Power of Maths)

12 Example 10: Expand (a - b) 4. Example 11: Expand (x + y).. Picking out terms General term: (r + 1)st term is n C r (x) n - r (y) r Example: The fifth term in (p q) 7 has a binomial coefficient of 7 C 4. Fifth term 7 C 4 (p) ( q) 4 5p q 4 Example: The fourth term in (x + y) 8 has a binomial coefficient of 8 C. Fourth term 8 C (x) 5 (y) 56(x 5 )y 179x 5 y The Dublin School of Grinds Page 9 Mills & Kelly (Power of Maths)

13 Example 1: In the expansion of (p + q) 7, what is the term with q 5? Example 1: What is the term with a 4 in the expansion of (a b) 9? 5 y Example 14: In x + 4, what is the term with y? For you to practice Exercise : page 40 The Dublin School of Grinds Page 10 Mills & Kelly (Power of Maths)

14 . Factorisation 1. Factorisation technique 1: HCF Factorise the following by taking out the highest common factor: 5x + 10y x y 10xy 7x(x y) y(x y). Factorisation technique : Grouping Example 15: Factorise ax - bx + ay - by by grouping. Example 16: Factorise x - 8y - + 1xy by grouping. Example 17: Factorise ax - bx - ay + by + a - b by grouping. The Dublin School of Grinds Page 11 Mills & Kelly (Power of Maths)

15 . Factorisation technique : Trinomials Example 18: Factorise x - 5x - 1. Rough work Rough work ( )( ) Example 19: Factorise 4x - 1x - 1. Rough work Rough work ( )( ) The Dublin School of Grinds Page 1 Mills & Kelly (Power of Maths)

16 4. Factorisation technique 4: Difference of two squares a - b (a - b)(a + b) Remember as: (First) (Second) (First Second)(First + Second) 5x - 4y ( 5x) -( y) ( 5x- y)( 5x+ y) 400x - 49y x - a b Example 0: Factorise (x + y) - z. 5. Factorisation technique 5: Difference and Sum of two cubes a + b (a + b)(a - ab + b ) a - b (a - b)(a + ab + b ) Remember as: (First) + (Second) (First + Second)((First) (First) (Second) + (Second) ) (First) (Second) (First Second)((First) + (First) (Second) + (Second) ) x + 8y ( x) + ( y) ( x+ y)( x - xy+ 4y ) 8x x - 64a b The Dublin School of Grinds Page 1 Mills & Kelly (Power of Maths)

17 Example 1: Factorise (a - 1) + (a + 1). 6. Factorisation technique 6: Harder factors Example : Factorise x + xy + y - 1. Example : Factorise 9a - 1ab + 4b - 5c. The Dublin School of Grinds Page 14 Mills & Kelly (Power of Maths)

18 Example 4: Factorise the following fully: (a) 1 x 4x (b) 7x 8y (c) a + 6ac 4ab 1bc (d) 154x 50x + 4 (e) (x + y) (x y) (f) 15a + 10a b For you to practice Exercise : page 41 The Dublin School of Grinds Page 15 Mills & Kelly (Power of Maths)

19 4. Algebraic Modelling Algebraic modelling means translating a problem, stated in words, into an algebraic expression. Guidelines for modelling 1. Identify the quantity to be modelled (cost, price, area, length, etc.) and give it a symbol.. Draw a diagram if appropriate (unless given).. Identify the number of variables (one or more) and if appropriate put them on the diagram. 4. Write the quantity to be modelled in terms of this (these) variable(s). Example 5: The breadth of a rectangular field is 0 m longer than its length x. Write down an expression, in terms of x, for: (a) the breadth b, (b) the perimeter P, (c) the area A. Example 6: A running track has two straights and two semicircular ends. If x m is the length of each straight and y m the radius of each semicircular end, (a) find an expression in terms of x and y for: (i) the perimeter P, (ii) the area A 1 of the rectangular region ABCD, (iii) the total area A. (b) If x 100 m and y 1 85 m, find the perimeter to the nearest metre. (c) If a runner has an average speed of 0 km/h, how long does it take the runner to complete one full circuit of the track? D x m Straight C s Semicircle y m y m y m y m Semicircle s A Straight x m B The Dublin School of Grinds Page 16 Mills & Kelly (Power of Maths)

20 For you to practice Exercise 4: page 4 The Dublin School of Grinds Page 17 Mills & Kelly (Power of Maths)

21 Chapter 4: Polynomial and Rational Expressions 1. Polynomial Expressions Types 1. Linear expressions Examples: x, x + In general: ax + b (Standard form). Quadratic Expressions Examples: x, x + 1, x - x + In general: ax + bx + c (Standard form). Cubic expressions Examples: x, x + 1, x + x + x + 1 In general: ax + bx + cx + d (Standard form) Example 1: State what type of expression is shown and write each expression in standard form. Type of polynomial Standard Form (a) - x (b) x + - x (c) 1 - x (d) x + x - + x Linear Linear Quadratic Example : Multiply (x + 1)(x - 5) writing your answer in standard form. First First First: Last Last Last: The Dublin School of Grinds Page 18 Mills & Kelly (Power of Maths)

22 Linear Quadratic Cubic Example : Multiply (x + 1)(4x - x - 5) writing your answer in standard form. First First First: Last Last Last: Linear Linear Linear Cubic Example 4: Multiply (5x + 1)(x - )(x - 1) writing your answer in standard form. First First First First: Last Last Last Last: The Dublin School of Grinds Page 19 Mills & Kelly (Power of Maths)

23 Example 5: Multiply out the following writing your answer in standard form. (a) (-x + k)(4x + l) (b) (x - kx + )(x + l). Identities (x - )(x - ) x - 5x + 6 When you multiply out the two brackets on the left you get an identical expression on the right. This is an identity. Identities are true for all values of the variable x. LHS RHS (x - )(x - ) x - 5x + 6 Try x 1: (1 - )(1 - ) (-)(-1) (1) - 5(1) Be clever. Try x : ( - )( - ) (0)(1) 0 Be clever. Try x : ( - )( - ) (-1)(0) 0 () - 5() () - 5() The Dublin School of Grinds Page 0 Mills & Kelly (Power of Maths)

24 Example 6: If 10x - 9x + (5x - k)(x - 1), find k. Example 7: If x + x + kx + (x - 1)(x + ax - ), find k and a. Method 1 (Lining up) Method (Choosing values) The Dublin School of Grinds Page 1 Mills & Kelly (Power of Maths)

25 Example 8: If x + 5x - 1 is a factor of x + 11x + 9x -, find the other factor. Example 9: If x - 1 and x + are factors of 6x + x - 18x + 8, find the other factor. Example 10: If x + is a factor of 9x - 19x - 10, find the quadratic factor. The Dublin School of Grinds Page Mills & Kelly (Power of Maths)

26 Example 11: If x + px - q is a factor of x + px - x + q, find the values of p and q. Example 1: If (x - 1) is a factor of x + ax - bx + 1, show that a (b - 5). The Dublin School of Grinds Page Mills & Kelly (Power of Maths)

27 Example 1: If x + bx - 5 is a factor of x - x + kx + 10, find the other factor and b and k. Example 14: ax + is a factor of 18x + kx + 6x -. Find k and a and the other factor. The Dublin School of Grinds Page 4 Mills & Kelly (Power of Maths)

28 Example 15: If (x - p) is a factor of x + qx + r, show: (i) r p, (ii) q -p. Example 16: If x - px + 1 is a factor of ax + bx + c, show c a(a - b). For you to practice Exercise 5: page 45 The Dublin School of Grinds Page 5 Mills & Kelly (Power of Maths)

29 . Division of polynomials Cubic Quadratic Cubic Linear, Quadratic, Quadratic Linear Linear Linear Always factorise first if you can. x -8 x x x x 4 x x 4 x- - x- ( - )( + + ) + + ( ) ( x-) x + x x x + 11x-14 x- 16x - 8x + 6x 8x- Example 17: Simplify 4x 4x - - x + 1. x + 1 For you to practice Exercise 6: page 46 The Dublin School of Grinds Page 6 Mills & Kelly (Power of Maths)

30 4. Rational Expressions A rational expression is one expression divided by another. Addition and subtraction Technique for adding and subtracting rationals Find the lowest common denominator (LCD) 1 1 xy x x y xy Example 18: Simplify (a) x x, (b) 4 7 x -1 - x -x- 5. Multiplication Technique for multiplying rationals Multiply the tops and multiply the bottoms and/or cancel. 4 -ab 15bc ab c c 5-4 bc 9ab abc a The Dublin School of Grinds Page 7 Mills & Kelly (Power of Maths)

31 Example 19: Simplify (a) a b - ab ab + a a- b, (b) x x x x. Division (Double-decker fractions) Technique for multiplying rationals Multiply above and below by the LCD of all fractions. Example 0: Simplify x x. + - x x For you to practice Exercise 7: page 47 The Dublin School of Grinds Page 8 Mills & Kelly (Power of Maths)

32 Chapter 5: Exponentials and Logs 1. Exponential Expressions (powers) An exponential expression can be written in the form a p where p R. a is called the base of the expression. p is called the power, the index or the exponent of the expression. 1. The multiplication rule Rule 1: a p a q a p + q Remember as: When you multiply two exponential expressions with the same base, you add the powers. x x 5 e x e x+ + 1 Formula and Tables Book: Page 1 (Indices and logs) 6 - x x 4 ( a+ b) ( a+ b). The division rule Rule : a a p q p a - q Formula and Tables Book: Page 1 (Indices and logs) Remember as: When you divide two exponential expressions with the same base, you subtract the power on the bottom from the power on the top. x x 6 6. The one rule ( a+ b) 0 x y Rule : a The power of a power rule Formula and Tables Book: Rule 4: (a p ) q a pq Page 1 (Indices and logs) Remember as: When you put an exponential expression to a power, you multiply the two powers. ( 6 5 ) ( x ) e x - ( ) 4 e x e y y x Formula and Tables Book: Page 1 (Indices and logs) The Dublin School of Grinds Page 9 Mills & Kelly (Power of Maths)

33 5. Powers of products and quotients Rule 5: ( ab) ab p p p p and a a b b p p Formula and Tables Book: Page 1 (Indices and logs) ( ab ) 4 5 xy z a 4 b 6. Negative powers Rule 6: a - p 1 1 and a p - p a a p Formula and Tables Book: Page 1 (Indices and logs) - - x - zy 1 - x ( xy) - ( ) a The flipping trick: b - p b a p x y Non-whole number powers 4 1 Rule 7: a 1 q q a Formula and Tables Book: Page 1 (Indices and logs) p q 1 q 1 p p q Other fractional powers: a ( a ) ( a ) The Dublin School of Grinds Page 0 Mills & Kelly (Power of Maths)

34 - Example 1: Simplify ( ), ( ) ( ) a b ( 5) 1 Example : Simplify ( x ) ( x e e ) x x e ( e ) , giving your answer in the form e ax + b. 6 t Example : The population P of yeast cells after t hours is given by P 10 ( ). t 6 (a) Show that P 10. (b) Find the population P 1 after n hours. (c) Find the population P after (n + ) hours. (d) Find the percentage change in population from n hours to (n + ) hours. Cont... The Dublin School of Grinds Page 1 Mills & Kelly (Power of Maths)

35 For you to practice Exercise 8: page 48. Surds A surd expression is an expression involving square roots of variable(s) or numbers that cannot be simplified into a rational expression. Examples of surds:, x, x-1 a a Simplifying surds Use the following results: ab a b and a b Simplify the following surds: a b x 1 4 The Dublin School of Grinds Page Mills & Kelly (Power of Maths)

36 Adding and subtracting surds Add and subtract like terms only: Write in their simplest form: x+ x + x x + ( ) Multiplying surds Write in their simplest form: ( + 5 ) x( x+ ) ( + )( 4-5 ) ( + ) ( x y) + ( a+ b)( a- b) ( a- b) is the conjugate surd of ( a+ b). Important result: ( a+ b)( a- b) ( a) -( b) a-b ( n+ 1- n)( n+ 1+ n) The Dublin School of Grinds Page Mills & Kelly (Power of Maths)

37 Division When you divide surds, the answer should never have a surd in the denominator. This process of getting rid of surds on the bottom is called rationalising the denominator. Write in their simplest form: Example 4: Rationalise the denominators of the following: 5 x+ y ( a), ( b). - x- y For you to practice Exercise 9: page 50 The Dublin School of Grinds Page 4 Mills & Kelly (Power of Maths)

38 . Logs A log is just another way to write a power. is the base. log 8 to base equals. 8 8 log 8? log 8? In general, log a x is the power to which you must put the base a to get x. log 9, log 7, log 16, log 1, log Escaping from logs (Hooshing)? or log a x y x y a 4 log log log log log log log Log Rules 1. The addition rule Rule 1: log x+ log y log ( xy) a a a Tip: You can only add logs with the same base when applying this rule. 1 log 6+ log + log 5 log ( ) log loga x + loga y+ loga log a ( ) log a. The subtraction rule Rule : log x- log y log a a a x y Tip: You can only subtract logs with the same base when applying this rule. log - log ( ) log log The Dublin School of Grinds Page 5 Mills & Kelly (Power of Maths)

39 log k 6( x ) 4 log k ( x ) log k log k( ) log k ( ) loga x+ loga y- loga z log a. Multiplication by a number rule log 5 log log k Rule : klog x log x a a 1 log 16 log log 4loga x- loga y loga - loga log a 4. Change of Base Rule 4: log a log x log b b x a log Change to base : log ( x- 1) log loge Change to base e: loga x log log8 Change to base 8: log16 8 log log e Tip: If you invert a log, you interchange the base with the expression inside the log. log a b 1 log a b Invert: log Invert: log 6 The Dublin School of Grinds Page 6 Mills & Kelly (Power of Maths)

40 All the rules together 1. Breaking logs down Break the following into individual logs: Technique Breaking down a log into a string of individual logs (a) Multiplication + logs (addition) (b) Division logs (subtraction) (c) Power Multiplies logs log ( xy ) log + log log + log 1- x log log ( ) x - log log ( ) - log x x 1 x + 1 log4 log log 4 4 y y y - log ( ) log 4 4 ( log 4 ( ) log4 ) - - log 4 ( ) log 4 Example 5: The magnitude M of an earthquake on the Richter scale is given by E M log 10, where E is the energy, in joules, released in the earthquake and E E J. 0 (a) Show that M [log10 E-log 10 E0]. (b) Evaluate log 10 E 0. (c) The 1906 San Francisco earthquake released J of energy. What was its magnitude on the Richter scale, if log ? Cont... The Dublin School of Grinds Page 7 Mills & Kelly (Power of Maths)

41 . Bringing logs together Technique Combining a string of individual logs into a single log (a) + Logs Multiply together on the top (b) Logs Multiply together on the bottom (c) A number multiplying a log moves into the log as a power Example 6: Evaluate log 4 8( x -1)- log 4( x+ 1) -log 4( x-1). For you to practice Exercise 10: page 5 For you to practice (Section ) Revision Questions: page 56 The Dublin School of Grinds Page 8 Mills & Kelly (Power of Maths)

42 Exercises EXERCISE 1 1. Multiply out and simplify the following: (a) (x + )(x + 5) (b) (x + 7)(x + ) (c) ( y + 5)( y + 8) (d) (x 5)(x 1) (e) x(x )(x + 4) (f) (x + x + 1)(x + 1) (g) (x x + 5)(x ) (h) (x 1)(x + 5x 7) (i) ( x + 5x 6)(1 x). Multiply out the following. The answers are a difference of two squares expression, which simplifies the process. (a) (x + )(x ) (b) (x 1)(x + 1) (c) (4x 1)(4x + 1) (d) (x + 1)(x 1) (e) (x )(x + ) (f) (4x y)(4x + y) (g) (x 5)(x + 5) (h) (x n )(x n + ). Multiply out the following perfect squares: (a) (x + ) (b) (x + ) (c) (x 4) (d) (5x 4) (e) (x 11) (f) (4y 5) (g) (ax b) (h) (a + 1) (a 1) 4. Multiply out and simplify the following: (a) (x )(x + 1)(x + ) (b) (x 1)(x + )(x + 1) (c) (x + )(x 1)(x ) (d) (x + 1) [Hint: (x + 1) (x + 1)] (e) (x 1) [Hint: (x 1) (x 1)] (f) (x )(x + 1)(x ) 5. (a) If a x and b x + 5, find the following, in terms of x: (i) a + b (iv) b (ii) a b (v) ab (iii) a (vi) a b (b) If p x x + 1 and q x + x, find the following, in terms of x: (i) p + q (iii) p + q (ii) p q (iv) q p (c) If p x + 5x 1 and q 4 x x + 5x 1, find the following, in terms of x: (i) p + q (iii) p q (ii) q p (iv) q xp 6. Find the values of the following expressions: (a) x + 11y if x and y 5 (b) 4x y + 8 if x 1, y 6 (c) 5x if x (d) 5x if x (e) (5x) if x (f) x y if x and y (g) x + 5x 7 if x 5 (h) 5x + 7x if x (i) (x + y) if x and y 4 (j) x y xy 7 if x and y (k) x y 5x + 7(x y) if x and y 0 5 The Dublin School of Grinds Page 9 Mills & Kelly (Power of Maths)

43 EXERCISE 1. Use your calculator to evaluate: 5 C 0, 10 C 0, 18 C 0, 6 C 0, 8 C 0. What is n C 0?. Use your calculator to evaluate: 5 C 5, 10 C 10, 18 C 18, 6 C 6, 8 C 8 and hence evaluate n C n. Using your calculator show: (a) 5 C 5 C (c) 7 C 7 C 4 (b) 10 C 4 10 C 6 (d) 18 C 5 18 C 1 Make a conclusion. 4. Expand out the following: (a) (x + 1) 4 (e) (x + y) (b) ( p + q) 8 (f) ( ) 6 5. (a) Find the fifth term in ( p + q) 8. (b) Find the fourth term in (x + y) 7. (c) Find the third term in (x + y) 6. (d) Find the sixth term in ( p q) Find the term with: (a) p in the expansion of ( p + q) 7 (b) q 5 in the expansion of ( p + q) 8 (c) (0 6) 4 in the expansion of ( ) 1 (d) (0 85) 5 in the expansion of ( ) 9 (e) p 6 in the expansion of ( p + q) 10 (f) q in the expansion of ( p + q) 8 (c) (q + p) 5 (g) ( ) 4 (d) (x y) 4 The Dublin School of Grinds Page 40 Mills & Kelly (Power of Maths)

44 EXERCISE 1. Factorise the following by taking out the highest common factor: (a) x + 9x 18 (b) 8a 16a b (c) 7x y 14x y (d) (x y) 5x(x y) (e) m(a + b) n(a + b). Factorise the following by grouping: (a) ax + ax + x + (b) ax bx + ay by (c) x 6 bx + b (d) x z x y + y z (e) x 8y + 1xy (f) 1 ax 14by + abx y. Factorise the following trinomials: (a) x + 5xy 14y (b) 10x + 1x (c) 7x xy + y (d) a + ab b (e) (a 1) + (a 1) 15 (f) 0x 17x + (g) b x + bxc + c (h) 4p 4p Factorise the following difference of two squares and simplify: (a) 4x 1 (b) 5x y (c) x a b (d) 4m 81n (e) (x + y) z (f) (x y) (x + y) (g) (x + 1) z (h) ( Yoke) (Thing) (i) (a + b) (a b) (j) Factorise the following sums and differences of two cubes: (a) x + 64 (f ) 15x 64a b (b) x + 7y (g) (x ) + 8 (c) 8x 7 (h) (x ) + (x + ) (d) y (i) x (1 y) (e) a b + c (j) (Thing) ( Yoke) 6. Factorise the following fully: (a) x 8 (b) 18a 8b (c) 6x + 15xy 9y The Dublin School of Grinds Page 41 Mills & Kelly (Power of Maths)

45 (d) x y + x y (e) 8 7x (f) 4x 4xy + 6y (g) x + 4xy y (h) (a + 1) 9 (i) 16(x 1) 4 (j) a 578 (k) x 4 (l) x y 7x (m) a b 4 8a b (n) x 54 (o) cos q + sin q 7. Factorise the following fully: (a) x + xy + y 81 (b) a + 6a + 9 b (c) a + 8ab + 16b c (d) a ab + b 16c (e) x 4 y 4 (f) x 6 y 6 (g) a ab + b a + b (h) x + x x (i) x 4 + x + 1 (j) a 4 a + ba b 8. (a) Derive x y (x y)(x + xy + y ) from x + y (x + y)(x xy + y ). (b) Expand out (a 4b)(a + b) + ab and hence factorise this expression. (c) Expand out (x + 1) + (x 1) (7x 1) and hence factorise this expression. (d) A is the square PQRS of side x. B is the square shown of side y. y S P D y B y Using Area A Area B Area C + Area D, show that x y (x y)(x + y). x C R Q x The Dublin School of Grinds Page 4 Mills & Kelly (Power of Maths)

46 EXERCISE 4 1. An entrepreneur bought x phones at 0 each and sold y of them at 98 each. By completing the table, find the entrepreneurʼs net profit, in terms of x and y. Number of phones Price/unit Sold Bought Total F E. An L-shaped flowerbed is shown with DC x m and CB y m. If [AB] is 1 m longer than [CD] and [ED] is m longer than [CB], find expressions for: (a) the perimeter P, (b) the area A of the flowerbed, in terms of x and y. A D x C B y. The product of a number x and the square of another number y is greater than the product of the number x squared and the number y. Find an expression for D, the difference between the bigger number and the smaller number, in terms of x and y. 4. A petty cash box contains 10x one cent coins, 10x two cent coins, 5x five cent coins, 0x ten cent coins, 15x twenty cent coins, 7x fifty cent coins, 5x one euro coins and x two euro coins. It also contains y five euro notes, y ten euro notes and one 0 euro note. Complete the table below and find an expression for the total amount A of cash in the box in cents. Cash type 1c c 5c 10c 0c 50c Number Value in cents 5. (a) For the rectangle shown, find an expression in terms of x and y: (i) for the perimeter P of the rectangle, (ii) for the area A of the rectangle. (b) If x 0 and y 0, what is the length of the perimeter? What is the area of the rectangle? Rectangle x m y m The Dublin School of Grinds Page 4 Mills & Kelly (Power of Maths)

47 6. A farmer constructs a fence around a field in the shape of a trapezium ABCD with [AD] parallel to [BC] and [AB] perpendicular to [BC]. A Fence B y m x m River Fence (a) If BC is 4 m longer than AD and DC is m longer than AB, find an expression for the length L of fencing in terms of x and y. (b) Find an expression for the area A enclosed by the fence in terms of x and y. (c) Find y. 7. (a) If x is a whole number, find an expression in terms of x and y for: (i) the next whole number, (ii) the sum of these two consecutive whole numbers. (b) Find the values of the sum of two consecutive whole numbers if x is the first number by copying and completing the following: Sum of two consecutive whole numbers x x x x 4 x 5 x 6 x 7 What conclusion can you make? 8. r is the radius of a circle, centre O, inscribed in a square ABCD. D C D C Find an expression in terms of r for: (a) the circumference of the circle, (b) the perimeter of the square, (c) the area of the circle, (d) the area of the square, (e) the area of the shaded region. 9. (a) A man can swim at m/s and walk at 1 5 m/s. If he takes x seconds to swim from A to B and y seconds to walk from B to D, find an expression for the length of the journey ABCD, in terms of x and y. B A Road C Water (b) A woman can swim at 1 8 m/s and walk at 1 m/s. If she takes (x + 0) seconds to swim from A to C and ( y 10) seconds to walk from C to D, find an expression for the length of the journey ACD, in terms of x and y. 10. ABCD is a rectangular frame with a picture inside, as shown. D A 1 m (x + ) m 1 m Picture 1 m 1 m x m Find an expression in terms of x for the area of: (a) rectangle ABCD, (b) the picture, (c) the border. C B D O r A B The Dublin School of Grinds Page 44 Mills & Kelly (Power of Maths)

48 EXERCISE 5 1. Simplify the following and give your answer in descending powers of x: (a) x 5 + x (b) 5x 7x 6 x + 1 (c) x + x 5x + 7x 5x + 1 (d) x + 5x 8x + 7x (e) x + 5x x + x x + 7x. Multiply out the following and give your answer in order of descending powers of x: (a) (x + 1)(x + k) (f) (x x + 1)(x k) (b) (x + 1)(k x + 1) (c) (x 1)(k x + ) (d) (1 x)(x k) (e) (x + x)(x + k) (g) (x 1)(k x 5) (h) (x 1)(x + k x ) (i) (x + 1)(x + k x ) (j) (x c)(x + k x + d). (a) If x 1 is a factor of x k x +, find k. (b) If x + is a factor of k x + 4x 6, find k. (c) If x 1 and x + are factors of ax + bx + c, find a, b and c. (d) If x is a factor of 4x 4x, find the other factor. (e) If 4x 1 is a factor of 8x + k x + 7, find the other factor and k. (f) If ax + bx + 8 (x 1)(5x k), find k, a and b. (g) Find the quadratic polynomial with factors: (i) 5x 1 and x (ii) 10x and x (iii) 5x 1 and 4x 6 (h) Find the quadratic polynomial with factors: (i) x +, x (ii) x + 1, x + 1 (iii) x +, x 4. (a) If x + x + is a factor of x + k x + lx +, find the other factor and k and l. (b) If x + bx 5 is a factor of x x + k x + 10, find the other factor and b and k. (c) ax + is a factor of 18x + k x + 6x. Find k and a and the other factor. (d) If x 1 is a factor of x + x k x + 4, find k R and the other factors. (e) If x is a factor of P (x) x x x k, find k and the quadratic factor of P (x). (f) If (x + 1) and (x ) are both factors of k x 6x + bx 6, find k, b R and the other factor. 5. (a) If (x p) is a factor of x + qx + r, show: (i) r p, (ii) q p. (b) If (x 1) is a factor of x + ax (a + 1) + 1, a > 0, find a R. (c) If x px + q is a factor of x + px + qx + r, show: (i) q p, (ii) r 8p. (d) If x a is a factor of f (x) x + x + px + q: (i) show p q a, (ii) write f (x) in form (x a )(x + r). (e) If x is a factor of P (x) 6 + x 4x + x, find the other factors of P (x). (f) If x a is a factor of x c, show c a. (g) If x px + 1 is a factor of ax + bx + c, show c a(a b). (h) If (x 1) is a factor of ax + bx + 1, find a and b. (i) If (x a) is a factor of x + rx + q, show: (i) r a (ii) q a (j) If x + ax + b is a factor of x + qx + rx + s show: (i) r b + a(q a) (ii) s b(q a) The Dublin School of Grinds Page 45 Mills & Kelly (Power of Maths)

49 EXERCISE 6 1. Simplify the following: _ (a) x x _ (b) x x x _ (c) 5 x + 15x 5x (d) x + x x + 1 _ (e) x + x x + 1. Simplify the following: _ (a) x + 7 x + 14x + 8 x + (b) x 4 x + 5 x + 1 _ (c) 4 x 11x + x _ (d) 10 x 1 x + 7x 0 x 5 _ (e) x 5x x 5 (f) 6 x 7 x + x _ x x + 1 _ (f) 15x + 11x 14 x (g) x 4 x 6 _ + x (h) x x + _ 16 x 8 x + 6x (i) 8x _ 4 x + 6 (j) x (g) x + x x 1 x + 1 _ (h) 6 x + x x x 1 (i) (j) 7 x 1 _ x 1 x 7x + 6 _ x + x. The volume of the box shown is given by V x + 7x + 7x +. x + 1 h x + 1 Find: (a) h in terms of x, (b) the surface area A, in terms of x. 4. If x 1 is a factor of x + kx 4x + 1, show that k. 5. If x is a factor of x + ax + bx + 6, show that b + a If x 5x + 11 is a factor of x x + ax +, show that a 4. The Dublin School of Grinds Page 46 Mills & Kelly (Power of Maths)

50 1. Simplify the following: _ 4 (a) p + _ p p (b) x 1 x _ x (c) x 1 _ x + 1 x (d) α β β α (e) _ x 18x 4x 1 4x (f) m 1 m (g) (h) (i) ( j) 1 n + n 1 + n n + n 1 n a + 1 4a + 8 _ a 1 _ x x 1 + _ x _ 7 5a 10 + a 5 a 0 x + x _ 4 x + 1 x + 1x _ 1 x + 5x + 6 EXERCISE 7 (h) ( a ) ( a + ) (i) ( a b ) ( j) ( x + y ) ( x (k) ( x 1 ) ( x 4y ) + 1 )(4x 8). Simplify the following: _ (a) 4 x x _ (b) x 4 x (f) (g) (c) 6x 18 _ 15 5x (d) 6 a + a a + a _ (e) 5 a + 17a 6 a _ 6 b 51b 7 b 9 4 x + x 4 x 7x + (h) x 4 y 4 x y x 6 y 6 (i) ( x y )(x xy + y ) (j) 4. Simplify the following: 6 x 7 x + x x x. Simplify the following: a (a) 5 b 8 b 4ab (b) x 1 9x _ ab (c) a 6ab _ 4 b ab 8ab _ (d) a + 8a + 15 a 9 (e) x 11x + 10 x 4 _ a a ab + 5b x + x 5 (f) x 11x + 1 x + 11x 1 _ x + 6x 7 x 5x + 4 (g) y + 7 y 9y y + y y + (a) x x (b) x _ x (c) (d) x _ x x 1 _ x _ x 1 x (e) (f) (g) + 1 x _ 1 x 1 x + _ 10 x 1 + x + _ 15 at at a t a t x The Dublin School of Grinds Page 47 Mills & Kelly (Power of Maths)

51 EXERCISE 8 1. Evaluate the following without using your calculator: (a) (b) 5 (c) 5 4 (d) 10 8 (e) 1 01 (f) ( 1) 4 (g) ( 1) 1 (h) ( ) 4 (i) ( ) 5 (j) ( 1 ) ) ( 1 (k) ( (l) ) (m) ( 4 5 ) (n) ( 1_ ) (o) ( _ 4 ) (p) ( 1_ ). Evaluate the following without using your calculator: (a) 1 (b) (c) 4 (d) 5 (e) 4 (f) ( 6) 1 (g) ( 1 4 ) (h) 1 (i) 1 _ (j) 1 (k) 1 _ 1 (l) 1 1 (m) ( 8) (n) ( ) (o) ( 1 ). Evaluate the following exactly without using your calculator: (a) 9 1 (b) 5 (c) 5 4 (d) 8 1 (e) 6 1 (f) 64 (g) (h) 16 (i) (j) 4 49 (k) 4 1 (l) ( ) (m) ( ) 4 5 (n) ( 8) 1 (o) 100 (p) ( 4 9 ) 1 1 (q) ( _ 8 7 ) 1 (r) ( _ 8 7 ) (s) ( _ 8 0 (t) 8 (u) ( 1 _ 7 ) 7 ) 1 (v) ( 1 _ 7 ) (w) ( 1 4 ) 1 (x) ( _ 5 16 ) (y) ( 1 ) ( ) 0 ( 16 _ 9 ) The Dublin School of Grinds Page 48 Mills & Kelly (Power of Maths)

52 4. Write: (a) 8 in the form p, p > 0 _ (b) 7 in the form p, p > 0 (c) 4 in the form 1 p, p > 0 _ (d) 49 7 in the form 7 p, p > 0 7 (e) 15 5 in the form 5 p, p > (f) 15 _ ( 10 ) in the form 10 p, p > 0 4 (g) ( 4 _ ) 1 p in the form q, p > 0, q > 0 _ (h) 16 in the form p, p > 0 8 (i) _ 7 in the form p, p > 0 9 (j) ( ) 5 1 _ ( 5 ) _ 9 in the form 5 p, p > Write in the form a p or 1 p, where p R, p > 0: a (a) a 7 a _ a (j) a (b) (a 7 ) (c) a 7 a (d) a 7 _ a (e) a (f) a a (g) (a a ) (h) a 14 _ a (i) a a (k) (l) (m) (n) (o) _ a a a a ( a ) a ( a ) 1 ( a ) 1 ( a ) 1 a a ( a ) ( a ) a 1 6. Simplify the following, giving all your answers with positive powers: (a) x (xy ) y (b) 5 x 6 x 4 15 x (c) 8 x 4 7 x (d) 6 x 7 x 5 (e) (f) 4 x 5 7 x x 70 x (xy )4 _ 5xy 7x y (g) _ x x x (h) x _ 4 5x x x (i) (xy ) xy 9( x y ) (j) ( x y _ z ) (k) 4 7 y 5 y 5x y 14xy (l) (m) (n) (o) (p) (q) (r) (s) (t) (a + b ) (a + b )6 _ (a + b ) xya b ab x y x _ ( ) ( x ) x x a _ y a _ y _ 4 x x 1 5 y _ 5 y x y 1 4 x y 16( x _ y ) 8 y x 7. The population P of a certain strain of bacteria is given by P 1500 t, where t is the time in hours. (a) Find the population P 1 after 4 hours. (b) Find the population P after 10 hours. (c) Compare these populations by dividing P by P The mass M of a radioactive material in grams (g) left after t hours is given by _ M 0 16 t. (a) Show that M 0 4t. (b) Compare the mass M left after (x + 7 ) hours with the mass M 1 left after (x + 6) hours by dividing M by M 1. The Dublin School of Grinds Page 49 Mills & Kelly (Power of Maths)

53 9. The mass M of a drug in milligrams (mg) in a person s body after t hours is given by M t (a) Show that M 0 t. (b) Find M after 1 hour. (c) Find the percentage change in M between t and (t + 1). 10. The value V in of an investment after t years is given by V 5000(1 08) t. (a) Compare the amount V after (t + ) years with the amount V 1 after t years, giving your answer to four decimal places. (b) Find the value of the investment after five years, correct to the nearest euro. (c) Find the value of the investment after eight years, correct to the nearest euro. EXERCISE 9 1. Write the following surds in their simplest form: _ (a) 1 (d) 51 _ (b) 7 (e) 8 9 (c) 110 (f) _ (g) 100 _ (h) 4 x _ (i) 8 x y 1 x (j) y _ 16 _ z (k) 61 9 x y The Dublin School of Grinds Page 50 Mills & Kelly (Power of Maths)

54 . Simplify the following surds: _ (a) (b) (c) x _ + y _ y + x 5 y (d) (e) _ (f) (g) (h) x + x x x (i) a a y + y a y + a y _ (j) x 4x 1 + x 5x 75. Multiply out and simplify the following surds: (a) ( ) (b) (c) 6 5 _ 18 (d) 6 ( ) (e) ( + 1 ) (f) ( _ 1)( + 1) (g) ab (a _ b + b a ) (h) ( 7 + _ 1 ) _ (i) ( )( 7 1 ) (j) ( x + y ) (k) ( _ x + + _ x + 1 )( _ x + _ x + 1 ) (l) x ( x ) + 5( x x) (m) 1 ( a + x a x ) (n) (x + )(x 1)(x ) 1 (o) ( x 1 + 1)( x 1) x 1 )( _ 1 x + 1 ) (q) 4 ( y + _ (p) ( _ 1 y ) ( 4 y _ y ) (r) ( x + _ x ) (s) ( x _ y _ y x ) 4. Rationalise each denominator: _ 1 (a) 5 _ (b) 1 _ (c) 4 (d) 5 (e) 6 (f) 7 _ x (g) y _ (h) x y 1 (i) + 1 _ 1 (j) (k) (l) (m) 5. If y, evaluate y 4y BCD is an isosceles triangle. Find: 4 + x h B D 8 x (a) the height h in terms of x, x x (n) + x (o) a a x 4 + x (b) the area of the triangle in terms of x. 7. If x + and y, find x + xy + y. 8. Show _ 1 x y y x. y 9. If V 0 h _ , show that V can be 5 h _ written as V h 4 5 h. C The Dublin School of Grinds Page 51 Mills & Kelly (Power of Maths)

55 10. Kepler showed that the periodic time P for a satellite to make a complete orbit around a planet of mass M at a distance r from the centre of the planet is given by P π r, where G is a constant. GM M r 1. The mass M of an object moving at a speed M 0 v is given by M _, where M 1 v 0 is the c mass at rest and c is the speed of light. (a) Show that M _ can be written as M M 0 c c v. c v _ (b) If v 4c 5, show that M 5M A simple pendulum consists of a mass M on the end of a string of length L moving back and forth as shown. (a) Write down the periodic times: (i) P 1 for a satellite at r R, (ii) P for a satellite at r 4R. _ (b) Show that P 1 1 P If x, y and z are consecutive terms in a geometric sequence, then y x z y. Show that the following numbers are consecutively in a geometric sequence: (a) 5, _ 5, 7 (b) a, _ ab, b M L The time to swing from right to left and back again is given by T π L g, where g is a constant. (a) Write down the time T 1 if L d. (b) Write down the time T if L d 4. (c) Show that T 1. T L M The Dublin School of Grinds Page 5 Mills & Kelly (Power of Maths)

56 EXERCISE Evaluate the following exactly: (a) log (f) log p ( 1 p ) (b) log 10 (10 5 ) _ (c) log 10 ( 1 10 ) (d) log e e (e) log k ( 1 k ) _ (g) log 5 5 (h) log 1 16 (i) log 4 (j) log Write the following logs in exponential (power) form: (a) log (b) log 4 16 (c) log 4 ( 1 _ 16 ) (d) log _ (e) log 16 ( 1 4 ) 1 (f) log 1 (g) log a 1 0 (h) log b a c (i) log _ (j) log 16 8 _ 4. Write the following as a single log (in the form log a x) and simplify, where possible: (a) log + log 5 (b) log a + log 4a (c) log e (1 x) + log e (1 + x) (d) log log 5 1_ 7 (e) log 5 y + log 5 y (f) log 4 (x 1) + log 4 (x + x + 1) (g) log e x + log e x 1 (h) log x ( x 4 y ) + log x ( y ) x (i) log k ( x _ 1 1) + log k ( x + 1 ) (j) log x x + log x x 4. Write the following as a single log (in the form log a x): (a) log 6 log (b) log a log a (c) log e (x 1) log e (x + 1) (d) log 5 7 log 5 ( 1 7 ) (e) log 4 (x 1) log 4 (x + x + 1) (f) log e x + 1 log e x 1 (g) log 9 (x + 6) log 9 (x + ) (h) log b (ba + b ) log b (a + b) (i) log x xy lo g x y (j) log a x log a ( 1 x ) 5. Write the following as a single log (in the form log a x): (a) log 5 (b) log 7 (c) 1_ log 4 5 (d) _ log 6 (e) log k y The Dublin School of Grinds Page 5 Mills & Kelly (Power of Maths)

57 (f) log 5 _ 1 + x (g) 1_ log z y (h) _ log 7 8 y (i) x log e 1_ (j) log 5 ( 1 y ) 6. (a) Write log 5 in base. (b) Write log 7 4 in base 5. (c) Write log a y in base b. (d) Write log 5 (x ) in base 7. (e) Write log 10 a in base e. (f) Write log 5 in base. (g) Write log 4 x in base x. (h) Write log x 5in base 5. (i) Write (j) Write _ 1 in base x. log x _ 1 in base x. log e x 7. (a) Write the following as a string of logs: (i) log a x y (ii) log (u v ) ( (iii) log x ) 5 p q (iv) log ( 7 x y ) 6 (v) log a ( 1 ) a (vi) log k (x + ) (x 1 ) (vii) log ( (x + 7 ) 7 4 x ) _ (viii) log 5 ( x 1 1 x + 1 ) (b) If a log 10 and b log 10, express the following in terms of a and b: (i) log 10 6 (ii) log 10 4 (iii) log 10 ( 9 8 ) (iv) log (v) log 10 0 (vi) log _ (vii) log 10 ( 16 7 ) (viii) log 10 ( ) (ix) log (x) log 1 8. (a) Write the following as a single log and hence, evaluate exactly, where possible: (i) log log 10 4 (ii) log 1 log 7 (iii) 1_ log log 10 5 log 10 7 (iv) log + log log 18 (b) Write the following as a single log: (i) (ii) (iii) log 5 x + log 5 y log k x + log k y 1_ 4 log k z 1_ log (x 1) _ log ( x + 1 ) (iv) 7{log (x + 5) + log log (4x 1)} (v) log 4 a + log 1 4 a + log 4 a 1_ (vi) 4 log a ( x + x + ) + 1_ 4 log _ a ( x + 1 x + ) 1_ (vii) {log e 9 log e (x + 1) + log e (x + )} (viii) x log e x 1_ log e x + e log e x 9. If log y log x +, show that y 4x. 10. If log e A + log e y log e y + D, show that y Ae D. 11. If f (x) log a x, show that f (x + h) f (x) (x + h) x log a ( 1 + h 1 h x ). 1. If log 5 y log 5 x log 5 (x + 1) + c, _ x show that y x c. The Dublin School of Grinds Page 54 Mills & Kelly (Power of Maths)

58 1. Use the rules of logs to show that: (a) log a a x x (b) a log a x x 14. Simplify the following: (a) log 7 log (b) 4 (c) log 5 5 log (d) (e) log x (x 8 ) (f) e 1 log e x Check the answers to (a), (b), (c) and (d) on your calculator. _ 15. (a) Using log b a log c a log c b, solve for log c a. Hence, simplify: log log 4 log 4 5 log 5 6 log 6 7 log 7 8 (b) Simplify: log log 4 log 8 log n 16. Show that: (a) log a (x + _ x 1 ) + log a (x _ x 1 ) 0 (b) log b ( x + _ x 1 ) + log b ( x _ x 1 ) The loudness L of a sound in decibels is given by L 10 log 10 ( I I 0 ), where I is the intensity of the sound in W m and I W m. (a) Find: (i) the loudness in decibels of normal conversation which has an intensity of I 10 7 W m 1, (ii) the loudness of amplified rock music which has an intensity of 10 1 W m. (b) If a sound has a loudness L 1 for intensity I 1 and loudness L for intensity 1000I 1, show that L L 1 0. _ Show that log x + _ 1 log x + _ 1 log 4 x + _ 1 log 5 x + _ 1 log 6 x _ 1 log 70 x. 19. Evaluate the following exactly: (a) log 54 log (b) 5 log log (c) (d) log 8 log If log 10 x (1 + p) and log 10 y (1 p), show that x y 100. The Dublin School of Grinds Page 55 Mills & Kelly (Power of Maths)

59 REVISION QUESTIONS 1. (a) Express 5 in the form _ a b 5, a, b N. + 5 (b) If x 1 is a factor of x + x + k x + 6, k Z, find k and the other factors. (c) If ax + b is a factor of ax + (b a)x + c, find the second factor and show that b c.. (a) If x 1 and y 1 +, express y x _ xy in the form a b, a Z, b N. (b) Simplify _ 4 x + 1x + 0 _ x + 14x (c) If a b c d, show that a b a + b c d. c + d. (a) If 5x 0x + 8 a(x + b) + c for all x, find a, b, c Z. ax (b) If _ b c _ by (c a) _ cz, show that (a b) 6ax + by + cz 0. (c) Express ( x form x n + 1 x n, where n N. ) x ( x + 1 x ) in the 4. (a) Show that x 5 x + _ 1, x, simplifies to x a constant. x (b) Express x 1 + x as a single fraction. x + 1 (c) (i) If a b(c x) 6x 7 x for all x R. Find a, b, c N. (ii) If x + x + 1 is a factor of x + ax x + b, find a, b Z. 5. (a) Expand ( p + q). (b) What is the term with q in the expansion of ( p + q) 7? (c) Show that p + q ( p + q) pq( p + q). 6. (a) Show that 1 + x p + p simplifies to a 1 + x constant. 6y (b) (i) Express x(x + 4y) _ as a single fraction. x (ii) If x is a factor of 4x kx +, find the other factor and k R. (c) If x ax is a factor of x 5x + bx + 9, find a, b N. 7. (a) If x , evaluate x 4 x exactly (b) (i) Evaluate exactly (ii) Simplify ( 1 + _ x ) _ 8x. (c) Show: (i) n n n _ (ii) _ (iii) 5 x 8 x 6 x x 1 8. The number of phones produced by a factory per week is given by N 1 50 x y, where x is the average number of workers that attend per week and y is the average number of hours worked by each worker per week. (a) Find the number of phones produced in a week in which the average attendance is 56 workers and the average number of hours worked is 6. (b) For another factory producing the same 1 phone N 0 x y. Find the number of phones produced by this factory in a week in which the average attendance is 56 workers and the average number of hours worked is 6. (c) Show that N _ 1 _ 5y N x. (d) Show that for a 40-hour week in each factory, an attendance of 100 workers in each will produce the same number of phones per week. 9. The stopping distances in metres (m) of a car 5 travelling at v km/h is given by S 1 v _ 0 for a wet 4 road and S v 6 for a dry road. (a) Copy and complete the table below. Give the values in the last two columns, correct to two decimal places, and make a conclusion. 5 v km/h v (b) Show that S 1 v S 1 4 v 10. S 1 S The Dublin School of Grinds Page 56 Mills & Kelly (Power of Maths)