If 1n a and 1n b are real numbers, then 1n a # 1n b = 1n ab.

Property Combining Radical Expressions: Products

If 1n a and 1n b are real numbers and b ≠ 0, then 1n a

1n b= 5n a

b .

Property Combining Radical Expressions: Quotients

TEKS (7)(G) Rewrite radical expressions that contain variables to equivalent forms.

TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Additional TEKS (1)(C), (1)(G)

TEKS FOCUS

•Rationalize the denominator – To rationalize the denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical.

•Simplest form of a radical expression – The simplest form of a radical expression with index n is one where there are no nth power factors in any radical and the denominators are all rationalized.

•Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated

•Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.

VOCABULARY

You can simplify a radical expression when the exponent of one factor of the radicand is a multiple of the radical’s index.

You can simplify the product of powers that have the same exponent. Similarly, you can simplify the product of radicals that have the same index.

ESSENTIAL UNDERSTANDING

Same Exponent Same Index

�2 � �3 � �2 � 322 � 32 � (2 � 3)2

43 � 53 � (4 � 5)3 �4 � �5 � �4 � 5 33 3

9-2 Multiplying and Dividing Radical Expressions

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Problem 3

Problem 2

Problem 1

Simplifying a Radical Expression

What is the simplest form of 23 54x5 ?

23 54x5 = 23 33 # 2 # x2 # x3 Find all perfect cube factors.

= 23 33x3 # 23 2x2 n1ab = n1a # n1b

= 3x23 2x2 Simplify.

Multiplying Radical Expressions

Can you simplify the product of the radical expressions? Explain.

A 13 6 # 12

No. The indexes are different. The property for combining radical expressions for products does not apply.

B 13 −4 # 13 2

Yes. 13 -4 # 13 2 = 13 -4(2) = 13 -8 = -2.

TEKS Process Standard (1)(G)

Simplifying a Product

What is the simplest form of 272x3y2 # 210xy 3 ?

TEKS Process Standard (1)(D)

272x3y2 # 210xy3 = 2(72x3y2)(10xy3)

= 2720x4y5

= 2122(5)(x2)2(y2)2y

= 2122(x2)2(y2)2 # 15y

= 12 ∣ x2y2 ∣ # 15y

= 12x2y215y

The simplest form is 12x2y2 15y.

You need to multiply the radicands and find the perfect square factors.

Now find square roots.Since 272x3y 2 and 210xy 3 must be real numbers, x and y are nonnegative, so no absolute value symbols are needed.

What allows you to use the property for multiplying radicals?The radicals must be real numbers. The indexes must be the same.

How do you know when you are done simplifying?You are done when the radicand contains no perfect cube factors.

394 Lesson 9-2 Multiplying and Dividing Radical Expressions

Problem 5

Problem 4

Dividing Radical Expressions

What is the simplest form of the quotient?

A 218x5

12x3

218x5

12x3 = 518x5

2x3

= 29x2

= 3x

B 23 162y5

13 3y2

23 162y5

133y2

= 53 162y5

3y2

= 23 54y3

= 23 27y3 # 13 2

= 23 33y3 # 13 2

= 3y23 2

Rationalizing the Denominator

Multiple Choice What is the simplest form of A35x2

12y2z ?

23 90x2yz2

6yz 135x2

1312y2z

523 x2yz2

yz 523 x2z

A35x2

12y2z= 23

5x2

23 22 # 3y2z The radicand in the denominator needs 2,

32, y, and z2 to make the factors perfect cubes.

= 23 5x2

13 22 # 3y2z# 23 2 # 32yz2

13 2 # 32yz2

Multiply the numerator and denominator

by 322 # 32yz2.

=23 90x2yz2

13 23 # 33y3z3

= 23 90x2yz2

2 # 3yz Simplify.

=23 90x2yz2

6yz

The correct answer is A.

Do you need to include absolute value symbols?No. Both the divisor and dividend already require that x be nonnegative.

How do you choose what to multiply by?Choose a cube root with a radicand that will make each factor of the radicand in the denominator a perfect cube.

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PRACTICE and APPLICATION EXERCISES

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M E W O RK

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Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Multiply, if possible. Then simplify.

1. 18 # 132 2. 13 4 # 13 16 3. 13 9 # 13 -81

4. 14 8 # 13 32 5. 1-5 # 15 6. 13 -5 # 13 -25

7. 13 9 # 13 -24 8. 13 -12 # 13 -18 9. 150 # 175

Multiply and simplify.

10. 23 6 # 23 16 11. 28y5 # 240y2 12. 28x5 # 13x

13. 24 81x5y4 # 4232x3y 14. 223 2xy2 # 23 4x2y5 15. 324 18a9 # 24 6ab2

Divide and simplify.

16. 150015

17. 248x 3

23xy2 18.

256x5y5

27xy

19. 23

250x7y3

232x2y

20. 23

48x 3y 2

236x

4y 21. 220ab

245a2b3

Rationalize the denominator of each expression.

22. 1x12

23. 1518x

24. 13 x13 2

25. 14 214 5

26. 23xy2

15xy3 27. 1

312ab3c2

1310a3bc

28. Apply Mathematics (1)(A) The formula t = 52sa shows the time t that any

vehicle takes to travel a distance s at a constant acceleration a, starting from rest. What is the difference in time between a car accelerating at 16 m>s2 and one accelerating at 25 m>s2 for a distance of 200 m?

29. The base of a triangle is 118 cm and its height is 18 cm. Find its area.

30. Use Representations to Communicate Mathematical Ideas (1)(E) The formula

F = mv2

r gives the centripetal force F of an object of mass m moving along a circle of radius r, where v is the tangential velocity of the object. Solve the formula for v. Rationalize the denominator.

31. Apply Mathematics (1)(A) The circular velocity v in miles per hour of a satellite

orbiting Earth is given by the formula v = 51.24 * 1012

r , where r is the distance in miles from the satellite to the center of Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than the velocity of a satellite orbiting at an altitude of 200 mi? (The radius of Earth is 3950 mi.)

32. a. Simplify 22 + 23175

by multiplying the numerator and denominator by 175.

b. Simplify the expression in (a) by multiplying by 13 instead of 175.

c. Explain how you would simplify 22 + 23198

.

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396 Lesson 9-2 Multiplying and Dividing Radical Expressions

TEXAS Test Practice

49. What is the simplified form of the expression 3118xy 2

if x and y are positive?

A. 12x2xy B.

12y2xy C.

254xy 2

2xy D. 227xy 2

2xy

50. What are the solutions, in simplest form, of the quadratic equation 3x 2 + 6x - 5 = 0?

F. -6 { 1966 G. -6 { i124

6 H. -3 { 2163 J. -3 { i16

3

51. A triangle has the dimensions shown. What is the height of a triangle with equal area but a base of 36?

A. h3 B. 2h

3 C. 2h D. 3h

52. Find the axis of symmetry of the graph of the function y = -2x 2 - 5x + 4. Show your work.

h

12

Simplify each expression. Rationalize all denominators.

33. 15 # 150 34. 13 4 # 13 80 35. 2x5y5 # 3 22x7y6

36. 25x4

12x2y3 37. 13 14

137x2y

38. 3211x3y

-2112x4y

39. The mass m of an object is 180 g and its volume V is 15 cm3. Use the formula D = m

V to find the density D of the object.

40. Explain Mathematical Ideas (1)(G) Does 2x 3 = 23 x 2 for all, some, or no values of x? Explain.

41. Evaluate Reasonableness (1)(B) Explain the error in this simplification of radical expressions.

Determine whether each expression is always, sometimes, or never a real number. Assume that x can be any real number.

42. 23 -x2 43. 2-x2 44. 1-x

Simplify each expression. Rationalize all denominators.

45. 5216x4y4 46. 523 8000 47. A6y-3

x-4

48. Create Representations to Communicate Mathematical Ideas (1)(E) When

2xayb is simplified, the result is 1x cy 3d , where c and d are positive integers.

Express a in terms of c, and b in terms of d.

–2 · –2(–8)–8 = 16= 4=

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