What is the difference between a^{m/n} and a^{n/m}?

The position of each number is critical. In a^{m/n}, the denominator tells you the root and the numerator tells you the power: a^{m/n} = \sqrt[n]{a^m}. In a^{n/m}, they swap positions: a^{n/m} = \sqrt[m]{a^n}. These are different values. For example, 4^{1/2} = \sqrt{4} = 2, but 4^{2/1} = 4^2 = 16.

Can fractional exponents be negative?

Yes. A negative fractional exponent combines the negative exponent rule with the fractional rule: a^{-m/n} = \frac{1}{a^{m/n}}. For example, 8^{-1/3} = \frac{1}{8^{1/3}} = \frac{1}{2}. The negative applies to the whole exponent, so first take the reciprocal, then apply the fractional exponent rule to the denominator.

How do I know whether to use radical notation or fractional exponent notation?

Both mean the same thing, so use whichever makes the problem easier. Fractional exponents are clearer when you are simplifying using exponent rules: x^{3/2} \cdot x^{1/2} = x^{3/2 + 1/2} = x^2 is faster than converting back and forth between radicals. Radical notation is often clearer in word problems or when the root itself is the focus.

Do fractional exponents work with negative bases?

Yes, but with a restriction: if the denominator is even (like a^{1/2} or a^{3/4}), negative bases do not produce real answers because no real number to an even power is negative. For example, (-4)^{1/2} is not real. If the denominator is odd (like a^{1/3}), negatives work fine: (-8)^{1/3} = -2 because (-2)^3 = -8.

Fractional Exponent

Convert between radicals and exponents: a fractional exponent means the denominator is the root and the numerator is the power on the inside.

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

Solve a problem with Fractional Exponent

Type the problem. The solver will use Fractional Exponent where Fractional Exponent is the right tool, and tell you when it is not.

What each symbol means

What Fractional Exponent takes

Symbol	Meaning
$a$	The base—the number being raised to the power or placed under the radical. The base must be non-negative if $n$ is even, or any real number if $n$ is odd.
$m$	The numerator of the fractional exponent, telling you the power to apply to the base after you take the root (or the power to apply before taking the root).
$n$	The denominator of the fractional exponent, telling you which root to take—the $n$ -th root, such as the square root ( $n = 2$ ) or cube root ( $n = 3$ ).

When to use it

When you need to rewrite a radical as an exponent or simplify expressions that mix roots and powers.

Level

Usually taught in: Algebra II

Worked examples

1. Simplify a fractional exponent with small integers

Problem

Simplify

8^{2/3}

$8^{2/3} = \sqrt[3]{8^2}$
Apply the rule: the denominator 3 becomes the root index, and the numerator 2 becomes the power inside the radical.
$= \sqrt[3]{64}$
Calculate $8^2 = 64$ .
$= 4$
Find the cube root: $4^3 = 64$ , so $\sqrt[3]{64} = 4$ .

Answer: $4$

Fractional exponents let you rewrite radicals using exponent notation. Here, $8^{2/3}$ means take the cube root of 8, then square it—or equivalently, square 8 first then take the cube root. Both orders give the same answer.

2. Simplify a negative fractional exponent

Problem

Simplify

27^{-2/3}

$27^{-2/3} = \frac{1}{27^{2/3}}$
A negative exponent means take the reciprocal of the base raised to the positive exponent.
$= \frac{1}{\sqrt[3]{27^2}}$
Apply the fractional exponent rule to $27^{2/3}$ : the denominator 3 is the cube root, and the numerator 2 is the power inside.
$= \frac{1}{\sqrt[3]{729}}$
Calculate $27^2 = 729$ .
$= \frac{1}{9}$
The cube root of 729 is 9 because $9^3 = 729$ , so $\sqrt[3]{729} = 9$ .

Answer: $\frac{1}{9}$

This example combines two rules: negative exponents (which flip to reciprocals) and fractional exponents. The negative exponent applies to the whole base, not just the result. Always handle the reciprocal first, then apply the fractional exponent rule to the denominator.

3. Apply fractional exponents in a real-world context

Problem

A coffee shop tracks its daily profit using the formula

P(d) = 100 \cdot d^{3/2}

where

d

is the day number. What is the profit on day 4?

$P(4) = 100 \cdot 4^{3/2}$
Substitute day $d = 4$ into the profit formula.
$= 100 \cdot \sqrt[2]{4^3}$
Rewrite $4^{3/2}$ using the fractional exponent rule: denominator 2 means square root, numerator 3 means the power inside.
$= 100 \cdot \sqrt{64}$
Calculate $4^3 = 64$ . Note: $\sqrt[2]{x}$ is the same as $\sqrt{x}$ (the square root).
$= 100 \cdot 8$
Find the square root: $\sqrt{64} = 8$ because $8^2 = 64$ .
$= 800$
Multiply to get the final answer.

Answer: $\text{The profit on day 4 is } \$800$

Real-world formulas often use fractional exponents to model growth or decay patterns. Converting $4^{3/2}$ to radical form makes it easier to calculate by hand: you can think of it as first taking the square root of 4, then cubing the result.

Common mistakes

Where Fractional Exponent usually goes wrong

The mistake	Why it is wrong	The fix
Writing $4^{3/2} = 4^3 \div 4^2$ or only computing the power part: $4^3 = 64$ without taking the root.	The numerator and denominator are part of a single exponent; they are not separate operations to apply or to divide.	Remember that $4^{3/2} = \sqrt[2]{4^3} = \sqrt{64} = 8$ . You must apply both the root (from the denominator) and the power (from the numerator) to get the correct answer.
Writing $8^{-1/3} = -\sqrt[3]{8} = -2$ or treating the negative sign as a negation of the answer.	A negative exponent means you take the reciprocal of the base raised to the positive exponent, not negate the final result.	$8^{-1/3} = \frac{1}{8^{1/3}} = \frac{1}{\sqrt[3]{8}} = \frac{1}{2}$ . First apply the reciprocal rule because of the negative sign, then apply the fractional exponent rule.
Reversing the fraction: writing $16^{1/4} = \sqrt[1]{16^4}$ or using the numerator as the root index.	The denominator of the fractional exponent is the root index, not the numerator. Using the numerator gives you a completely different (and wrong) value.	$16^{1/4} = \sqrt[4]{16^1} = \sqrt[4]{16} = 2$ . The denominator 4 tells you to take the fourth root; the numerator 1 means raise to the first power (so just $16$ under the radical).

Tips and when to use something else

The denominator of the fractional exponent is always the root you are taking: $a^{1/2}$ is a square root, $a^{1/3}$ is a cube root, and $a^{1/n}$ is an $n$ -th root.
You can compute $a^{m/n}$ in either order: $(\sqrt[n]{a})^m$ or $\sqrt[n]{a^m}$ both work and give the same answer, so choose whichever is easier to calculate with your numbers.
For negative bases, pay attention to the denominator: if $n$ is even (like $a^{1/2}$ or $a^{2/4}$ ), negative bases do not give real answers; if $n$ is odd (like $a^{1/3}$ ), negatives work fine.
When using the Product Rule for Exponents with fractional exponents, convert to a common denominator first: $a^{1/2} \cdot a^{1/3} = a^{3/6} \cdot a^{2/6} = a^{5/6}$ .

Related formulas

Product Rule for ExponentsThe Product Rule for Exponents simplifies multiplication of powers with the same base by adding the exponents into a single power.Power Rule for ExponentsSimplify nested exponents by multiplying the exponents together: this is the fundamental tool you need whenever one exponent is itself raised to a power.Negative ExponentA negative exponent tells you to take the reciprocal: a^{-n} equals 1/a^n. Use this to rewrite expressions with negative powers.Simplifying RadicalsBreak down square roots of products into simpler parts you can work with—essential for simplifying radical expressions in algebra.Square RootSquare root finds the non-negative number that, when multiplied by itself, gives your starting number; use it to undo squaring in equations.

Frequently asked questions

What is the difference between $a^{m/n}$ and $a^{n/m}$ ?: The position of each number is critical. In $a^{m/n}$ , the denominator tells you the root and the numerator tells you the power: $a^{m/n} = \sqrt[n]{a^m}$ . In $a^{n/m}$ , they swap positions: $a^{n/m} = \sqrt[m]{a^n}$ . These are different values. For example, $4^{1/2} = \sqrt{4} = 2$ , but $4^{2/1} = 4^2 = 16$ .
Can fractional exponents be negative?: Yes. A negative fractional exponent combines the negative exponent rule with the fractional rule: $a^{-m/n} = \frac{1}{a^{m/n}}$ . For example, $8^{-1/3} = \frac{1}{8^{1/3}} = \frac{1}{2}$ . The negative applies to the whole exponent, so first take the reciprocal, then apply the fractional exponent rule to the denominator.
How do I know whether to use radical notation or fractional exponent notation?: Both mean the same thing, so use whichever makes the problem easier. Fractional exponents are clearer when you are simplifying using exponent rules: $x^{3/2} \cdot x^{1/2} = x^{3/2 + 1/2} = x^2$ is faster than converting back and forth between radicals. Radical notation is often clearer in word problems or when the root itself is the focus.
Do fractional exponents work with negative bases?: Yes, but with a restriction: if the denominator is even (like $a^{1/2}$ or $a^{3/4}$ ), negative bases do not produce real answers because no real number to an even power is negative. For example, $(-4)^{1/2}$ is not real. If the denominator is odd (like $a^{1/3}$ ), negatives work fine: $(-8)^{1/3} = -2$ because $(-2)^3 = -8$ .

Need a different method?

The full solver is not scoped to one formula — type any problem and it will pick the method.

Open the math solver

Reviewed 2026-09-18