7-5
Rational Exponents
and Radicals
Vocabulary
Review
Write the simplified form of each square root expression.
1. 16
2. 100
4. Underline the correct word to complete the sentence.
3. 169
In the expression 23 , the number 3 is the base / exponent .
Vocabulary Builder
index (noun) in deks
Related Words: radical sign, radicand
4
Example: In the radical 16 , the index is 4, which means, “what number times
itself four times is 16?”
Use Your Vocabulary
Circle the index of each radical expression.
5
Write the cube of each number.
8. 23
4
3
5. 32
6. 27
9. 33
7. 256
10. 43
11. 53
For each statement, write T for True or F for False.
12.
13.
14.
Chapter 7
The index of 3x4 is 2.
The degree of a cube root is 4.
4
The expression 625 means, “what number times itself four times is 625?”
210
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Main Idea: The index of a radical gives the degree of the root, so the cube root has
an index of 3. If there is no index, the degree is 2.
Problem 1 Finding Roots
Got It? What is the simplified form of each expression?
3
5
3
b. 32
a. 27
3
c. 64
15. To evaluate 27 , you need to find
equal factors of
.
16. Write the missing factors of 27.
3
3
27 Z
3
17. What is the simplified form of 27 ?
Write the missing factors for each expression. Then write the simplified form.
5
5
18. 32 Z
5
32 3
19. 64 ;
<
3
64 Key Concept
1
3
Equivalence of Radicals and Rational Exponents
If the nth root of a is a real number and m and n are positive integers, then
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1
m
n
n
n
an a and a n am ( a )m .
Problem 2 Converting to Radical Form
Got It? What is each exponential expression in radical form?
5
a. a 6
2
1
b. 5x 3
c. (54y) 3
20. Underline the correct words to complete the sentence.
The numerator / denominator of the rational exponent is the index of the radical
and the numerator / denominator of the rational expression is the exponent
of the radicand.
Fill in each blank to make a true statement.
1
21. a 6 a
1
1
23. y 3 22. x 3 x
Write the index of each radical expression.
5
24. a 6 a5
1
25. 5x 3 5x
211
Lesson 7-5
2
26. Complete the steps to convert (54y) 3 to radical form.
2
2
(54y) 3 (
)3
27
Find at least one factor of 54y that is a cube.
(2y)
2
3
Use the power of a product property.
2
2
1
Rewrite 27 3 (27 3)2
(2y) 3
2
(2y) 3
3
Simplify the exponent.
2
Simplify 32 and write (2y) 3 in radical form.
9 Z(2y)
2
27. So, (54y) 3 in radical form is
.
Problem 3 Converting to Exponential Form
Got It? Write each radical expression in exponential form.
3
3
a. s2
b. 12 x4
c. (4y)5
4
d. 256a8
3
28. s2
3
29. 12 x4
30. 4(y)5
index
exponent of radicand
index
exponent of radicand
index
exponent of radicand
Underline the correct word to complete each sentence.
31. The index of a radical is the numerator/denominator of a rational exponent.
32. The exponent of the radicand is the numerator/denominator of a rational
exponent.
Write the numerator and denominator in exponential form for each expression.
3
3
33. s2 s
34. 12 x4 12x
36. What number multiplied by itself 4 times has a product of 256?
4
35. (4y)5 (4y)
37. Circle the expression that shows a8 written in exponential form.
8
a8
a4
4
a4
a8
4
38. How do you write 256a8 in exponential form?
Chapter 7
212
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Write the index and the exponent of the radicand for each radical expression.
Problem 4 Using a Radical Expression
Got It? You can estimate the metabolic rate of living organisms based on body
4
mass using Kleiber’s law. The formula R 73.3M3 relates metabolic rate R
measured in Calories per day to body mass M measured in kilograms. What is the
metabolic rate of a man with a body mass of 75 kg?
39. Write the number to complete each step.
4
R 73.3 Z
3
Substitute the man’s body mass into the formula for M.
Use a calculator to simplify.
40. To the nearest calorie, the metabolic rate is about
calories a day.
Lesson Check t Do you UNDERSTAND?
Write a rule for multiplying two radicals with the same radicand.
Justify why your rule works.
41. Circle the expression that could represent the multiplication of radicals with the
same base but a different index and a different exponent in the radicand.
n
c
a
(am)(ab)
n
a
(bm)(cn)
n
(am)(am)
42. Write each radical expression using exponents. Then multiply the expression.
c
n
am ab a
a
a
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43. Write a rule to explain your answer to Exercise 32. Justify why your rule works.
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Math Success
Check off the vocabulary words that you understand.
radicand
index
radical sign
Rate how well you can rewrite expressions involving radicals and rational exponents.
Need to
review
0
2
4
6
8
Now I
get it!
10
213
Lesson 7-5