Radical Expressions

1. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Rewrite radical expressions by using
rational exponents.
Simplify and evaluate radical expressions
and expressions containing rational
exponents.
Objectives

2. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
index
rational exponent
Vocabulary

3. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
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.
n
When a radical sign shows no index, it represents a
square root. Even roots MUST be a positive Answer.
Reading Math

4. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
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

5. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Method 1 Evaluate
the root first.
(4)1
Write with a radical.
64
1
3
4
Evaluate the root.
Evaluate the power.
Check It Out! Example 3a
Write the expression in radical form, and
simplify.
Method 2 Evaluate
the power first.
Write will a radical.
4
Evaluate the power.
Evaluate the root.
( )
1
3
64 ( )1
3
64
3
64

6. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Method 1 Evaluate
the root first.
(2)5
Write with a radical.
4
5
2
32
Evaluate the root.
Evaluate the power.
Check It Out! Example 3b
Write the expression in radical form, and
simplify.
Method 2 Evaluate
the power first.
Write with a radical.
32
Evaluate the power.
Evaluate the root.
( )
5
2
4 ( )5
2
4
2
1024

7. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Write each expression by using rational
exponents.
Simplify.
a. b.
3
4
81
10
3
Check It Out! Example 4
9
3
10
1000
Simplify.
5
1
2
c.
2
4
5

8. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Method 1 Evaluate
the root first.
(5)3
Write with a radical.
625
3
4
125
Evaluate the root.
Evaluate the power.
Check It Out! Example 3c
Write the expression in radical form, and
simplify.
Method 2 Evaluate
the power first.
Write with a radical.
125
Evaluate the power.
Evaluate the root.
( )
3
4
625 ( )3
4
625
4
244,140,625

9. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Simplify the expression. Assume that all
variables are positive.
Check It Out! Example 2a
Product Property.
Simplify.
Factor into perfect fourths.
2  x
2x
4 4
16x
4
24 •
4
x4
4
24 •x4

10. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Find all real roots.
a. fourth roots of –256
A negative number has no real fourth roots.
Check It Out! Example 1
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.

11. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
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. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
The properties of square roots in Lesson 1-3 also
apply to nth roots.

13. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents

14. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Check It Out! Example 2b
Quotient Property.
Product Property.
Rationalize the numerator.
Simplify.
Simplify the expression. Assume that all
variables are positive.
8
4
4
3
x
4
2
27
3
x

15. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Check It Out! Example 2c
3 3
7 2
x x
Product Property of Roots.
Simplify.
Simplify the expression. Assume that all
variables are positive.
x3
3 9
x

16. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Rational exponents have the same properties as
integer exponents (See Lesson 1-5)

17. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Product of Powers.
Simplify each expression.
Simplify.
Evaluate the Power.
6
Check It Out! Example 5a
Check Enter the expression
in a graphing calculator.

18. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Simplify each expression.
(–8)
– 1
3
Check It Out! Example 5b
1
–8
1
3
1
2
–
Negative Exponent Property.
Evaluate the Power.
Check Enter the
expression in a graphing
calculator.

19. Holt Algebra 2
3-2 Radical Expressions and Rational Exponents
Quotient of Powers.
Simplify each expression.
Simplify.
Evaluate the power.
5
2
Check It Out! Example 5c
25
Check Enter the expression
in a graphing calculator.