1. 8-6 Warm Up Lesson Presentation Lesson Quiz
Radical Expressions and Rational Exponents
Warm Up
Lesson Presentation
Lesson Quiz
Holt ALgebra2

2. Simplify each expression.
Warm Up
Simplify each expression.
1. 73 • 72
16,807 2.
118
116
121
3. (32)3
729 4. 75
5 3 20 7
2 35 7 5.

3. Objectives Rewrite radical expressions by using rational exponents.
Simplify and evaluate radical expressions and expressions containing rational exponents.

4. Vocabulary
index
rational exponent

5. You are probably familiar with finding the square root of a number
You are probably familiar with finding the square root of a number. These two operations are inverses of each other. Similarly, there are roots that correspond to larger powers.
5 and –5 are square roots of 25 because 52 = 25 and (–5)2 = 25
2 is the cube root of 8 because 23 = 8.
2 and –2 are fourth roots of 16 because 24 = 16 and (–2)4 = 16.
a is the nth root of b if an = b.

6. The nth root of a real number a can be written as the radical expression , where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root.

7. When a radical sign shows no index, it represents a square root.
Reading Math

8. Example 1: Finding Real Roots
Find all real roots.
A. sixth roots of 64
A positive number has two real sixth roots. Because 26 = 64 and (–2)6 = 64, the roots are 2 and –2.
B. cube roots of –216
A negative number has one real cube root. Because (–6)3 = –216, the root is –6.
C. fourth roots of –1024
A negative number has no real fourth roots.

9. Check It Out! Example 1
Find all real roots.
a. fourth roots of –256
A negative number has no real fourth roots.
b. sixth roots of 1
A positive number has two real sixth roots. Because 16 = 1 and (–1)6 = 1, the roots are 1 and –1.
c. cube roots of 125
A positive number has one real cube root. Because (5)3 = 125, the root is 5.

10. The properties of square roots in Lesson 1-3 also apply to nth roots.

11. When an expression contains a radical in the denominator, you must rationalize the denominator. To do so, rewrite the expression so that the denominator contains no radicals.
Remember!

12. Example 2A: Simplifying Radical Expressions
Simplify each expression. Assume that all variables are positive.
Factor into perfect fourths.
Product Property.
3  x  x  x
Simplify.
3x3

13. Example 2B: Simplifying Radical Expressions
Quotient Property.
Simplify the numerator.
Rationalize the numerator.
Product Property.
Simplify.

14. Simplify the expression. Assume that all variables are positive.
Check It Out! Example 2a
Simplify the expression. Assume that all variables are positive. 4 16 x 4
4 24 •4 x4
Factor into perfect fourths.
4 24 •x4
Product Property.
2  x
Simplify. 2x

15. Simplify the expression. Assume that all variables are positive.
Check It Out! Example 2b
Simplify the expression. Assume that all variables are positive. 8 4 3 x
Quotient Property.
Rationalize the numerator.
Product Property. 4 2 27 3 x
Simplify.

16. Simplify the expression. Assume that all variables are positive.
Check It Out! Example 2c
Simplify the expression. Assume that all variables are positive. 3 9 x
Product Property of Roots. x3
Simplify.

17. A rational exponent is an exponent that can be expressed as , where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents. m n

18. The denominator of a rational exponent becomes the index of the radical.
Writing Math

19. Example 3: Writing Expressions in Radical Form
Write the expression (–32) in radical form and simplify. 3 5
Method 1 Evaluate the root first.
Method 2 Evaluate the power first. ( ) - 3 5 32
Write with a radical.
Write with a radical.
(–2)3
Evaluate the root.
Evaluate the power. - 5
32,768 –8
Evaluate the power. –8
Evaluate the root.

20. ( ) ( ) Check It Out! Example 3a 641 3
Write the expression in radical form, and simplify.
Method 1 Evaluate the root first.
Method 2 Evaluate the power first. ( ) 1 3 64
Write with a radical. ( ) 1 3 64
Write will a radical.
(4)1
Evaluate the root.
Evaluate the power. 3 64 4
Evaluate the power. 4
Evaluate the root.

21. ( ) ( ) Check It Out! Example 3b 45 2
Write the expression in radical form, and simplify.
Method 1 Evaluate the root first.
Method 2 Evaluate the power first. ( ) 5 2 4
Write with a radical. ( ) 5 2 4
Write with a radical.
(2)5
Evaluate the root. 2
1024
Evaluate the power. 32
Evaluate the power. 32
Evaluate the root.

22. ( ) ( ) Check It Out! Example 3c 6254
Write the expression in radical form, and simplify.
Method 1 Evaluate the root first.
Method 2 Evaluate the power first. ( ) 3 4
625
Write with a radical. ( ) 3 4
625
Write with a radical.
(5)3
Evaluate the root. 4
244,140,625
Evaluate the power.
125
Evaluate the power.
125
Evaluate the root.

23. Example 4: Writing Expressions by Using Rational Exponents
Write each expression by using rational exponents. A. B. 4 8 13 = m n
a a = m n
a a 15 5 3 13 1 2
Simplify. 33
Simplify. 27

24. Write each expression by using rational exponents.
Check It Out! Example 4
Write each expression by using rational exponents. a. b. c. 3 4 81 9 3 10 2 4 5
103
Simplify. 5 1 2
Simplify.
1000

25. Rational exponents have the same properties as integer exponents (See Lesson 1-5)

26. Example 5A: Simplifying Expressions with Rational Exponents
Simplify each expression.
Product of Powers. 72
Simplify. 49
Evaluate the Power.
Check Enter the expression in a graphing calculator.

27. Example 5B: Simplifying Expressions with Rational Exponents
Simplify each expression.
Quotient of Powers.
Simplify.
Negative Exponent Property. 1 4
Evaluate the power.

28. Example 5B Continued
Check Enter the expression in a graphing calculator.

29. Check It Out! Example 5a
Simplify each expression.
Product of Powers.
Simplify. 6
Evaluate the Power.
Check Enter the expression in a graphing calculator.

30. Simplify each expression.
Check It Out! Example 5b
Simplify each expression.
(–8)– 1 3 1 –8 3
Negative Exponent Property. 1 2 –
Evaluate the Power.
Check Enter the expression in a graphing calculator.

31. Check It Out! Example 5c
Simplify each expression.
Quotient of Powers. 52
Simplify.
Evaluate the power. 25
Check Enter the expression in a graphing calculator.

32. Example 6: Chemistry Application
Radium-226 is a form of radioactive element that decays over time. An initial sample of radium-226 has a mass of 500 mg. The mass of radium-226 remaining from the initial sample after t years is given by To the nearest milligram, how much radium-226 would be left after 800 years?

33. Negative Exponent Property.
Example 6 Continued
800
1600
500 (200– ) = 500(2– ) t
Substitute 800 for t. 1 2
= 500(2– )
Simplify.
= 500( ) 1 2
Negative Exponent Property. =
500 2 1
Simplify.
Use a calculator.
≈ 354
The amount of radium-226 left after 800 years would be about 354 mg.

34. Check It Out! Music Application
To find the distance a fret should be place from the bridge on a guitar, multiply the length of the string by , where n is the number of notes higher that the string’s root note. Where should the fret be placed to produce the E note that is one octave higher on the E string (12 notes higher)?
Use 64 cm for the length of the string, and substitute 12 for n.
= 64(2–1)
Simplify.
Negative Exponent Property.
Simplify.
= 32
The fret should be placed 32 cm from the bridge.

35. ( ) Lesson Quiz: Part I Find all real roots. 1. fourth roots of 625
–5, 5
2. fifth roots of –243 –3
Simplify each expression. 3. 4.
256 y 8
4y2 4 2
5. Write (–216) in radical form and simplify. 2 3
= 36 ( ) - 2 3
216 5 3 21
6. Write using rational exponents. 3 5 21

36. Lesson Quiz: Part II
7. If $2000 is invested at 4% interest compounded monthly, the value of the investment after t years is given by What is the value of the investment after 3.5 years? $