Fractional Exponents | Moosa Academy

A fractional exponent looks unusual, but it is just a root written differently. Expressions such as $a^{1/2}$ , $a^{3/2}$ and $a^{2/5}$ carry two pieces of information: the denominator tells you which root to take, and the numerator tells you which power to raise the result to.

Case 1 When the numerator is 1

\[ a^{\frac{1}{n}} = \sqrt[n]{a} \]

With a numerator of 1, the denominator alone decides the order of the root. Nothing else happens — the expression is simply a root in disguise.

4^{\frac{1}{2}} = \sqrt{4} = 2

. A root sign with no number written on it means a square root.

8^{\frac{1}{3}} = \sqrt[3]{8} = 2

, since

2 \times 2 \times 2 = 8

16^{\frac{1}{4}} = \sqrt[4]{16} = 2

, since

2 \times 2 \times 2 \times 2 = 16

Each of these asks the same style of question: which number, multiplied by itself that many times, produces the value inside?

Case 2 When the numerator is not 1

\[ a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m \]

The work now happens in two steps, and the order matters for keeping the numbers small:

Taking the root first keeps the arithmetic manageable. Raising to the power first would work too, but the intermediate numbers grow quickly.

Example Three worked evaluations

Evaluate $4^{\frac{3}{2}}$ , $8^{\frac{2}{3}}$ and $16^{\frac{5}{4}}$ .

4^{\frac{3}{2}}

: first

\sqrt{4} = 2

, then

2^3 = 8

8^{\frac{2}{3}}

: first

\sqrt[3]{8} = 2

, then

2^2 = 4

16^{\frac{5}{4}}

: first

\sqrt[4]{16} = 2

, then

2^5 = 32

⟹ 8, 4 and 32

Notice how each root came out as 2, leaving only a small power to finish. That is the benefit of rooting before raising.

Note Two other ways to write it

The same quantity can be arranged in either of these forms, and all of them give identical results:

a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^m

a^{\frac{m}{n}} = \sqrt[n]{a^m}

Checking $8^{\frac{2}{3}}$ the second way: $8^2 = 64$ , and $\sqrt[3]{64} = 4$ — the same answer as before, but with a much larger number along the way.

Note The conversions at a glance

a^{\frac{1}{2}} = \sqrt{a}

a^{\frac{1}{3}} = \sqrt[3]{a}

a^{\frac{1}{4}} = \sqrt[4]{a}

a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m

Only two things need remembering: the denominator gives the root, and the numerator becomes the power applied afterwards. If the numerator is 1, the problem is purely a root.

Example Reading the fraction correctly

Evaluate $27^{\frac{2}{3}}$ .

The denominator is 3, so take the cube root:

\sqrt[3]{27} = 3

The numerator is 2, so square that result:

3^2 = 9

⟹ 27^(2/3) = 9

Swapping the two numbers would give $27^{\frac{3}{2}}$ , an entirely different quantity. The denominator is always the root, never the power.

Note Mistakes to avoid

Swapping numerator and denominator — the denominator gives the root.

Multiplying the base by the fraction, as if

4^{\frac{1}{2}}

meant

4 \times \frac{1}{2}

Raising to the power first and then struggling with a large number under the root.

Forgetting that a root sign without a number written on it means a square root.

Ignoring the numerator once the root has been taken, and stopping too early.

Summary

A fractional exponent is a root: the denominator gives the order of that root.
When the numerator is 1, a^(1/n) is simply the nth root of a.
When the numerator is m, take the root first and then raise the result to the power m.
a^(m/n) can also be written as (a^(1/n))^m or as the nth root of a^m — all give the same value.
Taking the root before the power keeps the numbers small and the arithmetic simple.