section 9.3 multiplying and dividing radical expressions

Section 9.3 Multiplying and Dividing Radical Expressions

9.3 Lecture Guide: Multiplying and Dividing Radical Expressions

Objective: Multiply radical expressions.

To multiply and divide some radical expressions, we use the properties:

Multiplying Radicals

n n nx y xy for 0x and 0y

n

nn

x x

yy for 0x and 0y

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

1. 32 2

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

2. 2 8x x

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

3. 3 23 27x x

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

4. 3 5 2 3

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

5. 4 5 3 5 7

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

6. 3 3 2 5 5 3

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

7. 8 7 1 2 7 3

Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.

8. 5 3 5 3y x y x

Simplify each expression. Assume x > 0 and y > 0.

9. 520x

Simplify each expression. Assume x > 0 and y > 0.

10. 3 728x y

Simplify each expression. Assume x > 0 and y > 0.

11. 4 83 16x y

Simplify each expression. Assume x > 0 and y > 0.

12. 5 103 54x y

Objective: Divide and simplify radical expressions.

Dividing Radicals

When dividing radical expressions, we do not always get a result that is rational. In this case, answers are typically given without any radicals in the denominator.

Recall that 2x = ____________ for 0.x In general, n nx _____________ for 0.x

As a warm-up to rationalizing the denominator of a radical expression, perform each multiplication.

13. 2 2

As a warm-up to rationalizing the denominator of a radical expression, perform each multiplication.

14. 8 2

As a warm-up to rationalizing the denominator of a radical expression, perform each multiplication.

15. 3 23 4 4

As a warm-up to rationalizing the denominator of a radical expression, perform each multiplication.

16. 3 2 32 4x x

Perform each division, and then express the quotient in rationalized form. Assume x > 0 and y > 0.

17.2

5

Perform each division, and then express the quotient in rationalized form. Assume x > 0 and y > 0.

18.3

7

Perform each division, and then express the quotient in rationalized form. Assume x > 0 and y > 0.

19.5

2

x

y

Perform each division, and then express the quotient in rationalized form. Assume x > 0 and y > 0.

20. 3

1

2

Perform each division, and then express the quotient in rationalized form. Assume x > 0 and y > 0.

21. 3

4

x

Perform each division, and then express the quotient in rationalized form. Assume x > 0 and y > 0.

22. 3 2

4

2x

Conjugates Radicals: The conjugate of x y is

.x y

Write the conjugate of each expression. Then multiply the expression by its conjugate.

Expression Conjugate Product

23.

24.

25.

Example:

2 32 3

2 3 2 3 4 9

2 3

1

3 5 2 7

3x y

2 3x y

Perform the indicated divisions by rationalizing the denominator and then simplifying. Assume that all variables represent positive real numbers.

26.16

7 3

Perform the indicated divisions by rationalizing the denominator and then simplifying. Assume that all variables represent positive real numbers.

27.8

3 5

Perform the indicated divisions by rationalizing the denominator and then simplifying. Assume that all variables represent positive real numbers.

28.3

2 5

Perform the indicated divisions by rationalizing the denominator and then simplifying. Assume that all variables represent positive real numbers.

29.4x

x y

Perform the indicated divisions by rationalizing the denominator and then simplifying. Assume that all variables represent positive real numbers.

30. 3

5 7x y

Perform the indicated divisions by rationalizing the denominator and then simplifying. Assume that all variables represent positive real numbers.

31.5 2

x

x y

Determine whether or not each value of x is a solution of the equation 2 2 4 0.x x

32. 2x

Determine whether or not each value of x is a solution of the equation 2 2 4 0.x x

33. 2x

Determine whether or not each value of x is a solution of the equation 2 2 4 0.x x

34. 1 5x

Determine whether or not each value of x is a solution of the equation 2 2 4 0.x x

35. 1 3x