1 / 7
Concept — Integer Base
9 × 9 × 9 × 9 × 9
=
9
"The exponent tells you how many times to write the base as a factor."
Concept — Fraction Base
13 × 13 × 13 × 13
=
(13)
=
13
=
3
The exponent applies to the entire base — including fractions.
Concept — Variable Base
x * x * x * x * x
=
x
This works for any base — numbers, fractions, or variables.
Vocabulary — Reading Exponents
b n base exponent (power)
“b to the power of n”or“b to the nth”
4⁵
"four to the fifth power"
x³
“x cubed”or“x to the third power”
(13)⁴
"one-third to the fourth power"
Root Trees — Completed Examples
√9 = ?
9 × 3 3
9 = 3 × 3,  so √9 = 3
∛27 = ?
27 × × 3 3 3
27 = 3 × 3 × 3,  so ∛27 = 3
Root Trees — What Goes in the Boxes?
⁴√16 = ?
16 4 × 4 √16 = 4 2 × 2 × 2 × 2 ⁴√16 = 2
16 = 4 × 4,  4 = 2 × 2,  so ⁴√16 = 2
⁵√7 = ?
7 × × × ×
What equal factor, multiplied 5 times, gives 7?
Roots & Fractional Exponents
Roots and fractional exponents are two ways to write the same number — they are always equal!
27⅓ × 27⅓ × 27⅓
27 × × 27⅓ 27⅓ 27⅓ fractional exp. fractional exp. fractional exp. 27⅓ × 27⅓ × 27⅓ = 27 ✓
All three factors are equal — each one is ∛27
Three names for ∛8
8 × × ∛8 2 8⅓ radical notation value fractional exp. ∛8 = 2 = 8⅓ · all equal 2, and 2 × 2 × 2 = 8 ✓
∛8 = 2 = 8⅓ — all the same number