( a b) 4 (2) Powers If a is a non-zero real number, and p and q are integers then:- a p a q. ( a p ) q = a p q a p. a q. = a 1 2

Transcription

1 Powers If a is a non-zero real number, and p and q are integers then: a p a q a q a ( p+ q) ( a p ) q a p q a p a ( p q) a p a a p q a p a q a p Note: denominator everywhere does not equal zero Examples ( 7 5 ) ( 3 6 ) 6 6 Problems simplify () a b 3 ( a b) 4 () 3 ( x y) 5 x y 4 chapter A0

2 Variables () Expand and simplify the expression Solution:- x ( x ) x + 3 4x 4 ( ) + 5x ( ) + x + 6x + 9 x 4x + 4 x 6x 9 + 5x + 5x 5x 5 () The volume of the sphere with radius r is given by the formula :- V 4 3 πr3 Find () the volume of the sphere that has a radius. cm. () the radius of the sphere that has a volume 3. cm 3 Solution () V () 3. ( ) 4 3 π πr3 r π cm 3 r π cm Problem π r h, where r is the The volume of a cylinder is given by the formv radius of the base and h is the height of the cylinder.find the height of the cylinder that has volume 7 m 3 and hr. chapter A0

3 Solving Equation Linear equations [ ax b + q ] such that a,b,q are variables and a 0 Example Solve the equation () x 5 7 solution x x 6 () 5x + x solution 5x x x 4 (b) Simultaneous Linear equations Example Solve the pair of equations: x + 3y 5 5x y 6 () substitution method From first equation x 5 3y substitute in the second equation: 5 5 3y y 6 5 5y 4y 3 y 3 so x () Elimination Method multiply the first equation by 5 and multiply the second equation by so 0x + 5y 5 0x 4y 3 ( ) ( ) 5y 4y 5 3 y 3 substitute in the first equation x chapter A0 3

4 Quadratic Equations [ ax + bx + c 0 ], a 0 Solving quadratic equation by formula are The solutions of the equation ax + bx + c 0 x b or + b 3 ac a Example Solve the following equations by formula () x 3x () x + 6x 5 solve, x Scientific notation Scientific notation takes the following form:- (a number between and 0, not including 0 ) x (a power of 0 ) Example Write (a) 53 (b) 0.35 in scientific form solution (a) 53.53x0 (b) x0 chapter A0 4

5 Task Type 3.6*.9 Using the palette of icons Before you can create a variable you need to know about the palette. The palette is the strip of icons running down the left-hand edge of the Mathcad window. Here is a picture of the first few icons, with one of them highlighted. then click on the ( real ) icon in the palette to see the answer. Click elsewhere on the page or press the [ ] key to finish. You should obtain the answer 0.44 Defining a variable A variable is a quantity which can vary ; it is represented by a symbol ( the variable name ). The Assign value to variable palette icon is used to define a variable. Variables greatly enhance what you can do in Mathcad. By defining them you can link equations together and store intermediate results for use in further calculations. Task Define the variable x to have value.5 by x :.5 Display the values of x, 4x and x... x.5 4 x 0 x 6.5 ( If x denotes the length of the side of a square, then the values of 4x and x give the perimeter and the area of the square. ) MATHCAD A0 page

6 Working with variables When working with variables ( and Mathcad in general ) you often have the choice of a mouse / click way or a keyboard way of doing things. You have already seen that clicking on the icon and typing have the same effect. In fact Mathcad provides keyboard alternatives for all the palette icons. Icon Key Define variable : Colon - given by [Shift]; Task pi... [Ctrl]p Hold down the control key, [Ctrl], and type p Find the area of the base A and the volume V of a cylinder of height h ( 8 ) whose ends are circles of radius r (.5 ). The variables r and h are defined for you. Definition for the radius r of the circular base r :.5 Definition for the height h of the cylinder... h : 8 Enter the expressions below, first to define A and V, and then to display them. Define a variable A to be the area of the base. Type A then click ; click then type *r^ ( Alternatively just type A:[Ctrl]p*r^ ) Define a variable V to be the volume of the cylinder, that is A times h. Now display the value of the area A and volume V You should obtain the answers A V MATHCAD A0 page

7 Simplifying algebraic expression Enter the expressions below ( a b + 3 c 4 d) 4 a + 3 b c + d To simplify the expression above, use the following procedure : ( ) Click anywhere on the expression above. Press 'Up Arrow' [ ] until the selection box surrounds the entire expression like this - Then choose Simplify from the Symbolic menu. After a short pause Mathcad should display... simplifies to 5 a 5 b + 5 c 5 d Simplify (b) 0 a b 5 a b (e) 4 ( ) 3 x + + x + 3 Expanding algebraic expressions The procedure to expand expressions in Mathcad ( x + ) x ( ) Click anywhere on the expression above. Press 'Up Arrow' [ ] until the selection box surrounds the entire expression, like this - Then choose Expand Expression from the Symbolic menu. After a short pause Mathcad should display... expands to x x MATHCAD A0 page 3

8 Factorising algebraic expressions To factorise expressions in Mathcad, you once again need to select the entire expression within the blue selection box, then choose Factor Expression from the Symbolic menu. Have a go at entering and then factorising the three expressions below for yourself. Help and answers are available at the bottom of the page. Don't forget to enter all the multiplication signs when constructing these expressions in Mathcad! Factorise Help x (i) a b ab (ii) x 4 (iii) x x + The following keystrokes can be used to enter the three expressions. a b a b Type a^[ ][ ]*b-a*b^ x 4 x^[ ][ ]-4 x + x^[ ][ ]-*x+ The multiplication signs must be entered between products in Mathcad ; for example, for 'two times x' type *x not x. Powers are given by the ^ key ( [Shift]6 ). The expressions also require careful use of the 'Up Arrow' key [ ]. To factorise an expression, click on it and press 'Up Arrow' [ ] until the selection box surrounds the entire expression, like this - Then choose Factor Expression from the Symbolic menu. Answers Here are the answers we obtained - a b a b x by factoring, yields a b x 4 by factoring, yields x ( b + a) ( ) ( x + ) x + by factoring, yields x ( ) MATHCAD A0 page 4