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CHAPTER 5 Exponents and Roots 105
Roots Expressed as Exponents
Roots can be written as exponents by using the following two properties.
This ability is useful in algebra and calculus.
p ffiffiffi 1=n
Property 1 n a ¼ a
The exponent is a fraction whose numerator is 1 and whose denominator is
the root.
Examples
p ffiffiffi 1=2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=3 1 1 1=2
x ¼ x 3 2x þ 1 ¼ð2x þ 1Þ p ffiffiffi ¼ ¼ x
x x 1=2
p ffiffiffi m p ffiffiffiffiffiffi m=n
m
n
Property 2 ð aÞ ¼ n a ¼ a (If n is even, a must be nonnegative.)
The exponent is a fraction whose numerator is the power and whose denomi-
nator is the root.
Examples
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p ffiffiffiffiffi p ffiffiffiffiffi
5 3 3=5 5 6 6=5 2 7 2 7=2
x ¼ x x ¼ x ð2x 1Þ ¼ð2x 1Þ
q ffiffiffiffiffiffiffiffiffiffiffiffiffi 15 15
3 2 2=3 3=2
ð12xÞ ¼ð12xÞ p ffiffiffiffiffi ¼ 3=2 ¼ 15x
x 3 x
Practice
p ffiffiffiffiffiffiffiffi
1: 14x ¼
3
2: p ffiffiffiffiffiffi ¼
2x
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5
3: 6 ðx þ 4Þ ¼
3x 5
4: p ffiffiffiffiffiffiffiffiffiffiffi ¼
x 5
