Quick answer
To raise a fraction to a power: apply the exponent to numerator and denominator separately. (a/b)ⁿ = aⁿ/bⁿ. For fractional exponents, p/q means the q-th root of the p-th power. Negative exponents flip the fraction first.
- Apply the exponent to both parts separately: (a/b)^n = a^n / b^n. Applying it only to the numerator is the most common mistake on this type of problem.
- Negative exponents mean reciprocal first: (a/b)^(-n) = (b/a)^n. Flip the fraction, then raise to the positive power.
- Fractional exponents are roots: 1/2 is the square root, 1/3 is the cube root, p/q is the q-th root of the p-th power - applied to numerator and denominator separately.
- Zero exponent always gives 1 for any non-zero fraction. This follows from a^n / a^n = a^(n-n) = a^0 = 1.
- Always simplify the result. After raising to a power, divide by the GCD. (6/10)^2 = 36/100 - divide by 4 to get 9/25.
How to use our fraction exponent calculator
The calculator has two inputs: the base fraction and the exponent fraction.
- Base fraction - the fraction you want to raise to a power, e.g. 3/4. Enter 3 in the numerator field and 4 in the denominator field.
- Exponent - how many times to multiply the base by itself. An exponent of 2 means (3/4) × (3/4). Enter whole exponents as a fraction too: for the exponent 3, enter 3 in the top field and 1 in the bottom field.
The calculator returns the simplified result, the decimal equivalent, and step-by-step workings so you can see how the answer was reached.
Fractional exponent explained to a beginner
Think of (3/4)^2 as simply (3/4) × (3/4) - the fraction multiplied by itself. Multiply the tops: 3 × 3 = 9. Multiply the bottoms: 4 × 4 = 16. Result: 9/16.
Raising a proper fraction to a positive integer power always shrinks it, because you are taking a fraction of a fraction. Three-quarters of three-quarters must be less than three-quarters.
A fractional exponent like (4/9)^(1/2) runs the question in reverse: "what fraction, when squared, gives 4/9?" The answer is 2/3, because (2/3)^2 = 4/9. That is the entire meaning of the exponent 1/2 - it undoes squaring.
For (8/27)^(2/3): first find the cube root of each part (2/3), then square the result (4/9). Power and root, applied in two steps.
Positive whole exponent
The rule: raise both numerator and denominator to the same power.
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
$$\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$$
This works because (a/b)^n means multiplying the fraction by itself n times: (a/b)×(a/b)×…×(a/b). The numerators multiply to give a^n and the denominators multiply to give b^n.
Negative exponent
A negative exponent means take the reciprocal, then apply the positive exponent:
$$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}$$
$$\left(\frac{2}{3}\right)^{-1} = \frac{3}{2}$$
(2/3)^−1 = 3/2. (3/4)^−2 = (4/3)^2 = 16/9. The negative sign in the exponent always means "flip the fraction first."
Fractional exponent (roots)
A fractional exponent p/q combines a power (p) and a root (q):
$$\left(\frac{a}{b}\right)^{p/q} = \sqrt[q]{\left(\frac{a}{b}\right)^p} = \frac{\sqrt[q]{a^p}}{\sqrt[q]{b^p}}$$
The exponent 1/2 is the square root, 1/3 is the cube root, and so on:
$$\left(\frac{a}{b}\right)^{1/2} = \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$
$$\left(\frac{4}{9}\right)^{1/2} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3}$$
For a combined power and root:
$$\left(\frac{8}{27}\right)^{2/3} = \left(\frac{\sqrt[3]{8}}{\sqrt[3]{27}}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$$
(8/27)^(2/3): find the cube root of each part first (2/3), then square: 4/9.
| Fractional exponent | Meaning | Example |
|---|---|---|
| 1/2 | Square root | (4/9)^(1/2) = 2/3 |
| 1/3 | Cube root | (8/27)^(1/3) = 2/3 |
| 1/n | n-th root | (a/b)^(1/n) = ⁿ√(a/b) |
| 2/3 | Cube root, then square | (8/27)^(2/3) = 4/9 |
| 3/2 | Square root, then cube | (4/9)^(3/2) = 8/27 |
The symmetry in that table is worth noting: (4/9)^(3/2) = 8/27, and working backwards with (8/27)^(2/3) returns 4/9 exactly.
I use this as a quick verification whenever I am not sure about a fractional exponent result. If (a/b)^(p/q) = c/d, then (c/d)^(q/p) should return a/b - if it does not, something went wrong in the root or power step.
Zero exponent
Any non-zero fraction raised to the power 0 equals 1:
$$\left(\frac{a}{b}\right)^0 = 1 \quad (a \neq 0)$$
This follows from the exponent rule: a^n / a^n = 1 = a^(n−n) = a^0.
Worked examples for calculating fraction exponent
Example 1: (2/5)^3
Raise each part to the 3rd power: 2^3 = 8, 5^3 = 125. Result: 8/125. Decimal: 0.064.
Example 2: (3/4)^−2
Negative exponent → flip: (4/3)^2 = 16/9 = 1.777…
Example 3: (1/4)^(1/2)
Square root: √1/√4 = 1/2. Decimal: 0.5.
Example 4: (27/8)^(2/3)
Cube root first: ∛27/∛8 = 3/2. Then square: (3/2)^2 = 9/4 = 2.25.
Common mistakes when working out fraction exponent
Applying the exponent only to the numerator
(3/4)^2 ≠ 9/4. The exponent applies to both parts: 3^2 = 9 and 4^2 = 16, so (3/4)^2 = 9/16.
This error is persistent because it echoes what happens when multiplying a fraction by a whole number: 3/4 × 2 = 6/4 (numerator only changes).
Exponentiation is different - (3/4)^2 means (3/4) × (3/4), not 3 × 2 / 4. Keeping that distinction clear eliminates this mistake. A fast check: the denominator must also change when you raise to any power other than 1 or 0.
Misinterpreting a fractional exponent as a fraction
(4/9)^(1/2) is NOT the same as (4/9) × (1/2) = 2/9. The exponent 1/2 means square root: √4/√9 = 2/3.
Forgetting to simplify the result
After raising to a power, always simplify the resulting fraction with the GCD. (6/10)^2 = 36/100 → GCD = 4 → 9/25.
Frequently asked questions
How do you raise a fraction to a power?
Apply the exponent to both numerator and denominator separately. (a/b)^n = a^n / b^n. Simplify the result.
What does a fractional exponent mean?
p/q means raise to the p-th power and take the q-th root. (a/b)^(1/2) = square root, (a/b)^(1/3) = cube root.
What is (2/3)^3?
2^3 / 3^3 = 8/27 ≈ 0.296.
What is a negative fraction exponent?
Take the reciprocal first: (a/b)^(−n) = (b/a)^n. Example: (3/4)^(−2) = (4/3)^2 = 16/9.
Quiz: how well do you know fraction exponents?
1. What is the correct formula for raising a fraction (a/b) to a positive whole power n?
2. What is (27/8)^(2/3)?
3. What does a negative exponent like (3/4)^(-2) tell you to do first?
4. According to the fractional exponent reference table on this page, what does the exponent 2/3 mean?
5. A student calculates (3/4)^2 = 9/4. What mistake did they make?