We know what 2³ means (2 · 2 · 2 = 8) and what 2⁻¹ means (1/2). But what does 9^(1/2), "nine to the power of one half", mean? You can't multiply nine by itself "half a time".
Mathematicians agreed on a meaning that keeps all the familiar laws of indices working. And it turns out that a fraction in the index is another way of writing a root: 9^(1/2) = √9 = 3. (The index is the small raised number, also called the power or exponent. In the US, fractional indices are called rational exponents, and surds are called radicals.)
When you multiply powers with the same base, you add the indices: aʳ · aˢ = a^(r + s). If this is to work for halves too, then
9^(1/2) · 9^(1/2) = 9^(1/2 + 1/2) = 9¹ = 9.
So we are looking for a positive number that gives 9 when multiplied by itself. That is exactly √9 = 3. In the same way, 8^(1/3) · 8^(1/3) · 8^(1/3) = 8, so 8^(1/3) = ∛8 = 2.
The general rule
For a > 0 and whole numbers m, n (n at least 1) the rule is:
a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
We read it as: "a to the power m over n is the n-th root of a to the power m".
The denominator n tells you which root to take.
The numerator m tells you which power to raise to.
For example, a^(1/2) = √a, a^(1/3) = ∛a, a^(3/4) = ∜(a³).
Working out the value
💡 When working out a value, it is easier to take the root first, then raise to the power. The numbers stay small.
Worked example 1
8^(2/3)
Denominator 3, so take the cube root: ∛8 = 2.
Numerator 2, so square it: 2² = 4.
8^(2/3) = 4.
Worked example 2
32^(2/5)
⁵√32 = 2, because 2⁵ = 32.
2² = 4.
Worked example 3
36^(3/2)
√36 = 6.
6³ = 216.
Negative indices and fractions as the base
A negative index means "one over" (the reciprocal): a⁻ʳ = 1/aʳ.
Worked example 4
16^(−3/4)
16^(−3/4) = 1/16^(3/4).
∜16 = 2 and 2³ = 8.
16^(−3/4) = 1/8.
Worked example 5
(4/9)^(3/2)
Take the root of the numerator and the denominator: √4 = 2, √9 = 3.
(2/3)³ = 8/27.
Worked example 6
(1/8)^(−2/3)
A negative index turns the fraction upside down: (1/8)^(−2/3) = 8^(2/3).
8^(2/3) = 4 (see example 1).
Writing a root as a power
The rule works the other way round too: we can write a root as a power with a fraction in the index. We assume a > 0.
Root
√a
∛(a²)
⁵√(a³)
1/√a
1/∛(a²)
Power
a^(1/2)
a^(2/3)
a^(3/5)
a^(−1/2)
a^(−2/3)
Calculating with powers
The same laws of indices work as for whole-number indices:
Law
Example
aʳ · aˢ = a^(r + s)
a^(1/2) · a^(1/3) = a^(5/6)
aʳ ÷ aˢ = a^(r − s)
a^(5/4) ÷ a^(1/2) = a^(3/4)
(aʳ)ˢ = a^(r · s)
(a^(2/3))^(3/4) = a^(1/2)
Worked example 7
√a · ∛a
Write as powers: a^(1/2) · a^(1/3).
Add the indices: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
√a · ∛a = a^(5/6) = ⁶√(a⁵).
Worked example 8
√(a · ∛a)
Inside the root: a · ∛a = a¹ · a^(1/3) = a^(4/3).
A square root means the index 1/2: (a^(4/3))^(1/2) = a^(4/3 · 1/2) = a^(2/3).
√(a · ∛a) = a^(2/3).
Common mistakes
Treating the fractional index as multiplication. 8^(2/3) is not 8 · 2/3. The correct answer is (∛8)² = 4.
Swapping the numerator and the denominator. ∛(a²) = a^(2/3), not a^(3/2). The index of the root goes in the denominator.
Treating a negative index as a negative number. 16^(−3/4) = 1/8, not −8.
Adding fractions wrongly. 1/2 + 1/3 is not 2/5, it is 5/6.
Summary
a^(m/n) = ⁿ√(aᵐ): the denominator tells you which root, the numerator which power.
When working out a value, take the root first, then the power.
A negative index means "one over" (the reciprocal).
Multiplying powers: add the indices. Dividing: subtract them. Power of a power: multiply them.
