Chapter 7 – Powers,
Roots, and Radicals
7.1 – nth Roots and Rational Exponents
7.1 – nth Roots and Rational
Exponents
 Today we will learn how to:
 Evaluate nth roots of real numbers
7.1 – nth Roots and Rational
Exponents
 You can extend the concept of square roots to
cube and other roots.
 b is the square root of a if b2 = a
 b is the cube root of a if b3 = a
 b is the fourth root of a if b4 = a
 b is the nth root of a if bn = a
 You can write the nth root of a as n√a, where a is
a real number and n is the index of the radical
7.1 – nth Roots and Rational
Exponents
 Let n be an integer greater than 1, and let a be a
real number.
n
odd
even
a
any real number
Number of real nth roots
of a
one
Examples
3√8
=
3√-1 =
greater than 0
two
=
-4√16 =
0
one
√0 =
4√0 =
less than 0
none
4√16
4√-20,
no real
root
7.1 – nth Roots and Rational
Exponents
 Example 1
 Find the indicated nth root(s) of a.
 n = 5, a = -32
 n = 3, a = 27
7.1 – nth Roots and Rational
Exponents
 Example 2
 Solve the equation.
 6x3 = 384
 (x – 8)5 = 32
7.1 – nth Roots and Rational
Exponents
HOMEWORK
7.1 Worksheet
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7.1 – nth Roots and Rational
Exponents
 Rational Exponents
 You do not have to use a
radical to write an nth
root.
 Notice that the
denominator of each
rational exponent is equal
to the index of the
radical.
 You can write an nth root
of a using a fraction as an
exponent.
 √3 = 31/2
 3√7 = 71/3
 n√a = a1/n
7.1 – nth Roots and Rational
Exponents
 Example 3
 Evaluate the expression.
 1251/3
 161/2
 811/4
 (-64)1/6
7.1 – nth Roots and Rational
Exponents
 You can also use a fraction as an exponent to
write a combination of a power and an nth root.
 (3√2)4 = (21/3)4 = 24/3
 (5√9)7 =
7.1 – nth Roots and Rational
Exponents
 Rational Exponents
 Let a1/n be an nth root of a (a > 0 and a ≠ 1), and let
m be a positive integer.
 am/n = (a1/n)m = (n√a)m
 a-m/n = 1/am/n =
1
(a1/n)m
= 1
(n√a)m
7.1 – nth Roots and Rational
Exponents
 Example 4
 Rewrite (4√11)3 using rational exponents.
 Rewrite 1 using rational exponents.
5√7
 Rewrite 53/5 using radicals.
 Rewrite 3-2/5 using radicals.
7.1 – nth Roots and Rational
Exponents
 Example 5
 Evaluate the expression.
 1252/3
 8-4/3
7.1 – nth Roots and Rational
Exponents
HOMEWORK
7.1 Worksheet
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