Topic 16 - Radicals 1. Definition of a Radicals Definition: In Mathematics an efficient way of representing repeated multiplication is by using the notation a n, such terms are called exponentials where a is called the base and n the exponent. In general we say that a n is a to the power of n but in particular when the exponent is 2 we often use the word squared and when the exponent is 3 we use the word cubed. For example, n terms Here are some examples of exponentials with whole numbers as powers. 5 3 125 (called 5 cubed) 10 2 100 (called 10 squared) 2 5 32 (called 2 to the power of 5) Definition: A radical is a special version of an exponential; a radical is an exponential with a fractional power such as that cannot be simplified to a exponential with a integer power. For example,, are all examples of radicals. While,, are not truly radicals since they can be simplified to,, It can also be shown that the radical so a to the power of is the same as the nth root of a. Also the more generalized version of this property is that There are three basic operations involving all exponentials including radicals they are as follows. Multiplication Rule Division Rule Powers Rule Page 1
Example 1: Which of the following are radicals? (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Solution: (a) This has a fractional power that can t be simplified and so is a radical. (b) This has a fractional power that can t be simplified and so is a radical. (c) 5 This is not a radical. (d) This has a fractional power that can t be simplified and so is a radical. (e) This has a fractional power that can t be simplified and so is a radical. (f) (g) (h) This has a fractional power that can t be simplified and so is a radical. This is not a radical. This has a fractional power that can t be simplified and so is a radical. (i) 2 This is not a radical. (j) This has a fractional power that can t be simplified and so is a radical. (k) This is not a radical. (l) This is not a radical. Exercise 1: Which of the following are radicals? (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Page 2
2. Simplifying Numerical Radicals A numerical radical is a radical that has a number as its base. For example numerical radicals since they have bases of 8, 32 and 16 2 256 respectively while numerical radicals. are all are not Definition: A radical is in its simplest form when it is written as a multiple of a radical with the smallest base. The method we will use to simplify numerical radicals involves splitting the number into a product of two whole numbers where one of the numbers is the largest perfect square and then simplifying the resulting terms by using the property that for any numbers x and y In order to use this method we need to know the perfect squares and cubes, etc. Perfect Squares 1 4 8 16 25 36 49 64. Perfect Cubes 1 8 27 64 125 216 343 256. For example the 6 th perfect cube is 6 3 216 while the 8 th perfect square is 8 2 64 Example 1: Solution: The can be written as and as we choose the form as 36 is the largest perfect square we can use. We then simplify the result as follows. 6 ( since ) We can now say that the simplified form of is The process of simplifying a numerical radical that is a cube root is to split the number into a product of two whole numbers where one of the numbers is the largest perfect cube and then simplifying the resulting terms by using the property that for any numbers x and y Example 2: Solution: The can be written as and as we choose the form as 64 is the largest perfect cube we can use. We then simplify the result as follows. 4 ( since ) We can now say that the simplified form of is Page 3
Example 3: Solution: 2 ( since ) We can now say that the simplified form of is Example 4: Solution: 5 ( since ) We can now say that the simplified form of is Example 5: Solution: 4 ( since ) We can now say that the simplified form of is Example 6: Solution: 10 ( since ) We can now say that the simplified form of is Example 7: Solution: 3 ( since ) We can now say that the simplified form of is Exercise 2: Simplify the following numerical radicals. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Page 4
3. Simplifying Radicals Expressions The method used to simplify radicals expressions that contain variables involves expressing the radical as an exponential with a rational exponent we then express the rational exponent as a mixed fraction and simplify the result using the property that x a+b x a x b where a will be a whole number and b will be a fraction. Example 1: Solution: Express 3 x 3 x 1/2 x 3 Using the property that x 1/2 Example 2: Solution: Express 3 c 3 c 2/3 c 3 Using the property that x 2/3 Example 3: Solution: Express 2 b 2 b 3/4 b 2 Using the property that b 3/4 Example 4: Solution: Using the property that Expressing 1, 3 and 3 Using the property that x 1/2 Rearranging the terms Using the property that Page 5
Example 5: Solution: Using the property that Expressing Expressing 3 and 2 Using the property that x 1/2 Rearranging the terms Using the property that Example 6: Solution: Using the property that Expressing Expressing 2 and 3 Using the property that x m/n Rearranging the terms Using the property that Exercise 3 Simplify the following radical expressions. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) Page 6
4. Adding and Subtracting Radicals To add or subtract radicals we simplify the radicals using the mehod from secction 3 first and then we collect like terms as in algebra we then finally add or subtract the like terms. Example 1: Add and subtract the following radical expressions. (a) (b) (c) (d) + (e) + Solution (a): + + s Add like terms Solution (b): s Subtract like terms Solution (c): s Subtract like terms Solution (d): + 2 + 5 ) + ) + + Solution (e): + + + + + s Collect like terms Page 7
Example 2: Add and subtract the following radical expressions. (a) (c) (b) (d) Solution (a): Solution (b): s Add like terms s 14x Subract like terms Solution (c): Solution (d): s Add like terms s Add like terms Page 8
1. Add and subtract the following radicals. (a) (b) (c) + (d) + (e) + (f) (g) (h) (i) Exercise 4 Page 9
5. Multiplying Radicals To multiply radicals we do the following steps. Step 1: Multiply the expression using the distributive law or FOIL and the property Step 2: Simplify each of the radical terms. Step 3: Collect like terms (if necessary) Step 4: Add or subtract like terms (if necessary) Example 1: Multiply the following radicals. (a) (b) (c) (e) (g) (d) (f) (h) Solution (a): Using he distributive law Using the property 16 4 12 Solution (b): Using he distributive law Using the property Simplifying the radicals Solution (c): Using the property s Page 10
Solution (d): Using the property s Solution (e): Use F.O.I.L. s Add and subtract like terms Solution (f): Use F.O.I.L. 6b s 6b Add and subtract like terms Solution (g): s Add and subtract like terms Solution (h): s Page 11
Exercise 5 1. Multiply the following radicals. (a) (b) (c) (d) (e) (f) (g) (h) Page 12
To divide radicals we do the following steps. 6. Dividing Radicals Step 1: Divide the two radicals using the property Step 2: Simplify each of the radical terms. Example 1: Divide the following radicals. (a) (b) (c) (d) Solution (a): Using the property Simplifying the radical Solution (b): Using the property Simplifying the radical Solution (c): Using the property Simplifying the radical Solution (d): Using the property Simplifying the radical Page 13
There is a special case of division of radicals it is called Rationalizing the denominator it normally involves a fraction where the denominator does not simplify into the numerator. In these si8tuations we use a technique of multiplying the numerator and the denominator by the denominator of thye fraction. Example 2: Rationalize the denominator of the radical Solution: Multiply the fraction by the special version of 1 Simplify the fraction Example 3: Rationalize the denominator of the radical Solution: Multiply the fraction by the special version of 1 Simplify the fraction Example 4: Rationalize the denominator of the radical Solution: Multiply the fraction by the special version of 1 Simplify the fraction Example 5: Rationalize the denominator of the radical Solution:. Multiply the fraction by the special version of 1 Page 14
Exercise 6 1. Divide the following radicals. (a) (b) (c) (c) 2. Rationalize the denominators of the following radicals. (a) (b) (c) (d) Page 15
7. Solving Radical Equations Definition: A radical equation is one that contains a radical typically a or a These are examples of radical equations. x 4 3 5 2x 3 11 2x 17 x + 1 When you solve a radical equation you are trying to find usable value(s) of x that satisfy both sides of the equation. The main method used in solving a radical equation is to remove the radical by squaring or cubing both sides of the equation. The new equation formed can then be solved in the usual manner. It is important to check that your solutions work as it is possible to get unusable solutions. For example we can try to solve the radical equation we would discover that x 4 is a possible solution. However if you check this solution by substituting it into the original radivcal equation you will discover that it can t be used since So x 4 is not a usable solution as there is no real number that is equal to When these situations occur we cannot use x 4 as a solution and this may mean that there is no real solutions to the radical equation. The x 4 is sometimes called a phantom solution as it does not really exist. Example 1: Solve the radical equation x 4 Solution: x 4 x 16 Squaring both sides Check: Substitute x 16 into x 4 4 It works Page 16
Example 2: Solve the radical equation 3 Solution: 3 2x + 1 27 Cubing both sides 2x 26 Subtract 1 from both sides x 13 Divide both sides by 2 Check: Substitute x 13 into 3 3 3 It works Example 3: Solve the radical equation 5 Solution: 5 3x + 4 25 Squaring both sides 3x 21 Subtract 4 from both sides x 7 Divide both sides by 3 Check: Substitute x 7 into 5 5 5 It works Example 4: Solve the radical equation 2x 3 11 Solution: 2x 3 11 2x 3 121 Squaring both sides 2x 124 Adding 3 to both sides x 62 Divide both sides by 2 Check: Substitute x 62 into 2x 3 11 2(62) 3 11 11 It works Page 17
Example 5: Solve the radical equation 2x 17 x + 1 Solution: 2x 17 x + 1 2x + 17 (x +1) 2 2x + 17 (x + 1)(x + 1) 2x + 17 x 2 + 2x + 1 17 x 2 + 1 16 x 2 Solutions are x 4 and x 4 Check: Substitute x 4 into 2x 17 x + 1 2(4) 17 4 + 1 5 It works Check: Substitute x 4 into 2x 17 x + 1 2( 4) 17 4 + 1 3 This value of x does not work since is positive 3 and not 3 This means that this radical equation has only one real solution x 4 1. Solve the following radical equations. Exercise 7 (a) x 7 (b) 4 (c) 5 (d) 2x 5 9 (e) x + 1 (f) 5x 5 x + 1 (g) 2x + 7 (h) Page 18
8. Applications Using Radicals Example 1: Pythagoras Theorem. What is the length of the hypotenuse of the right angled triangle shown opposite. 10 x Solution: Pythogaras theorem states a 2 + b 2 c 2 where c is the hypotenuse. 24 This can be rearranged to give c 26 Example 2: Pythagoras Theorem. What is the length of the hypotenuse of the right angled triangle shown opposite. x + 4 2x + 6 Solution: Pythogaras theorem states a 2 + b 2 c 2 where c is the hypotenuse. 4x This can be rearranged to give c 2x + 6 2x + 6 2x + 6 (2x + 6) 2 4x 2 + 24x + 36 0 0 (13 +10)(x 2) Solutions x 2 and x Check x 2 The right angled triangle formed when x 2 is triangle shown opposite. This triangle exists and so x 2 is a usable solution with a missing hypotenuse of 10 units. 6 8 10 Check x The right angled triangle formed when x is triangle shown opposite. This triangle does not exist and so x Not a usable solution. is Page 19
So the solution to this problem is that x 2 and in this situation the hypotenuse 10. Example 3: The weight of an object in pounds is given by the formula W where L is the length of the object in inches. (a) What is the weight of an object if its length is 10 inches? (b) If the weight of the object is 10 pounds what is its length? Solution (a): W W W W 5 The weight of the object is 5 pounds. Solution (b): W 10 100 2L + 5 95 2L L The length of the object is 47.5 in Example 4: The period of a pendulum T in seconds is given by the formula T 6.28 Where L is the length of the pendulum chain in feet. If the period of a pendulum is 10 seconds how long is L the length of the pendulum chain? Solution: T 6.28 10 6.28 1.592 Divide both sides by 6.28 (1.592) 2 Square both sides 2.534464 (2.534464)9.8 L Multiply both sides by 9.8 24.8 L The pendulum chain will be 24.8 feet long!!! Page 20
Exercise 8 1. What is the length of the hypotenuse of the right angled triangle shown opposite? 7 x 10 2. What is the length of the hypotenuse of the right angled triangle shown opposite? x 2x + 3 2x+2 3. The weight of an object in pounds is given by the formula W where L is the length of the object in inches. (a) What is the weight of an object if its length is 7 inches? (b) If the weight of the object is 4 pounds what is its length? 4. The period of a pendulum T in seconds is given by the formula T 6.28 Where L is the length of the pendulum chain in feet. (a) (b) If the length of the pendulum is 19.6 feet what will be the period of the pendulum? If the period of a pendulum is 12.56 seconds how long is L the length of the pendulum chain? Page 21
Solutions Exercise 1: (a) not a radical (b) radical (c) radical (d) radical (e) not a radical (f) radical (g) not a radical (h) radical (i) radical (j) radical (k) radical (l) not a radical Exercise 2: (a) (b) (c) 2 (d) (e) (f) (g) 5 (h) 2 (i) (j) 3 (k) 2 (l) Exercise 3 (a) (b) (c) (d) ) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) Exercise 4: (a) (b) (c) (d) (e) (f) (g) (h) (i) Exercise 5: (a) (b) (c) (d) (e) (f) (g) x y (h) 6x 2y Exercise 6: 1.(a) (b) 2x (c) (d) 2.(a) (b) (c) (d) Exercise 7: (a) x 49 (b) x 3 (c) x 3 (d) x 43 (e) x 2 only (f) x 4 and x 1 (g) x 2 (h) x 1 Exercise 8: 1. Hypotenuse x 2. x 5 Hypotenuse 2x + 3 13 3.(a) Weight 3.(b) length 5.5 inches 4.(a) T Period of pendulum 8.88 sec 4.(b) L length of pendulum 39.2 feet Page 22