Fractional Exponent Calculator
Enter a base and a fractional exponent (numerator over denominator) to get the result, the root and power steps and the exact or radical form.
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Fractional Exponent Calculator
Our Fractional Exponent Calculator raises a base to a fractional (rational) power such as 2/3, 3/2 or −1/2. It turns the exponent into a root and a power, shows each step, gives the exact answer when there is one (82/3 = 4) and the simplest radical form when there is not (81/2 = 2√2). It also handles negative exponents, negative bases, decimals and fraction bases such as 1/4.
Quick answer
A fractional exponent m/n means take the n-th root, then raise it to the power m: xm/n = (ⁿ√x)m = ⁿ√(xm). For example, 82/3 = (³√8)² = 2² = 4.
- Denominator: the root (2 = square root, 3 = cube root)
- Numerator: the power
- Negative exponent: 1 over the result, so 16−3/4 = 1/8
- Negative base: real only when the reduced denominator is odd
What Is a Fractional Exponent?
A fractional exponent (also called a rational exponent) is a power written as a fraction. It is another way to write a root:
x^(1/3) = ³√x
x^(m/n) = (ⁿ√x)^m
The denominator says which root to take and the numerator says which power to raise it to. This works because (x1/n)n = x1 = x, which is exactly what an n-th root does.
How to Solve Fractional Exponents
- Reduce the fraction in the exponent if you can (4/6 becomes 2/3).
- Take the root given by the denominator.
- Raise it to the power given by the numerator.
- If the exponent is negative, take 1 over the answer.
= 3²
= 9
Taking the root first keeps the numbers small. Powering first gives the same answer, ³√(27²) = ³√729 = 9, but with bigger numbers along the way.
Fractional Exponent Rules
| Rule | Example |
|---|---|
Rootx^(1/n) = ⁿ√x | 9^(1/2) = 3 |
Root and powerx^(m/n) = (ⁿ√x)^m | 32^(3/5) = 8 |
Negativex^(−a) = 1/x^a | 4^(−1/2) = 1/2 |
Productx^a·x^b = x^(a+b) | √2·√2 = 2 |
Power of a power(x^a)^b = x^(ab) | (√4)³ = 8 |
Quotientx^a/x^b = x^(a−b) | 8^(2/3)÷8^(1/3) = 2 |
These are the ordinary exponent rules; they hold for fractional exponents whenever the base is positive.
Negative Fractional Exponents
A minus sign in the exponent means "1 over". Work out the positive exponent, then flip it:
= 1/(⁴√16)³
= 1/2³
= 1/8 = 0.125
A negative exponent never makes the answer negative. For a positive base it only makes it smaller than 1 (or larger, if the base is below 1).
Negative Bases
Odd roots of negative numbers are real, even roots are not:
(−8)^(2/3) = (−2)² = 4
(−4)^(1/2) = no real answer
The calculator reduces the exponent first, so (−8)2/6 is treated as (−8)1/3 = −2. Also note that −81/3 written without brackets usually means −(81/3) = −2; type the base as −8 to raise the negative number itself.
Simplest Radical Form
When the root is not a whole number, the calculator gives the decimal and, for whole-number bases, the simplest radical form. It pulls perfect powers out of the root:
= 2√2 ≈ 2.828427
12^(3/2) = √(12³) = √1728
= 24√3 ≈ 41.569219
Common Fractional Exponents
| Expression | Value |
|---|---|
| 4^(1/2) | 2 |
| 8^(1/3) | 2 |
| 16^(1/4) | 2 |
| 8^(2/3) | 4 |
| 9^(3/2) | 27 |
| 81^(3/4) | 27 |
| 125^(−2/3) | 1/25 = 0.04 |
| 2^(1/2) | ≈ 1.414214 |
| 2^(1/3) | ≈ 1.259921 |
Common Mistakes
- Mixing up the parts. In 82/3 the 3 is the root and the 2 is the power, not the other way round.
- Multiplying by the fraction. 82/3 is not 8 × 2/3 = 5.33. It is 4.
- Making the answer negative. 16−1/2 is 1/4, not −4.
- Even roots of negatives. (−9)1/2 has no real value.
- Calculator brackets. On most calculators type 8^(2/3), with brackets; 8^2/3 means 64 ÷ 3.
Key Terms
- Base
- The number being raised to a power.
- Exponent
- The power; here a fraction m/n.
- Radical
- A root sign, such as √ or ³√.
- Index
- The small number on a root sign; the exponent's denominator.
- Radicand
- The number under the root sign.
How to Use the Fractional Exponent Calculator
- Enter the base. Type the number being raised, such as 8. Negative numbers, decimals and fractions such as 1/4 all work.
- Enter the fractional exponent. Type the numerator (the power) on top and the denominator (the root) below, such as 2 over 3, or tap a common exponent such as 1/2 or 2/3. Use a minus sign on the numerator for a negative exponent.
- Read and copy the result. The answer appears as you type, with the radical form, the root step, the power step, the exact fraction or simplest radical and the decimal. Press Copy to copy it, or Clear or Esc to start again.
Example: enter 8 as the base, 2 on top and 3 below. The result is 4, with the radical form ³√(8²) = (³√8)², the root step ³√8 = 2 and the power step 2² = 4.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is a fractional exponent?
A fractional exponent is a power written as a fraction, such as x2/3. The denominator is a root and the numerator is a power, so xm/n = (ⁿ√x)m.
How do you solve fractional exponents?
Take the root given by the denominator, then raise the result to the power given by the numerator. 272/3 = (³√27)² = 3² = 9.
What is 8 to the power of 2/3?
82/3 = 4. The cube root of 8 is 2, and 2² = 4.
What does an exponent of 1/2 mean?
An exponent of 1/2 means the square root. 91/2 = √9 = 3. An exponent of 1/3 means the cube root, so 81/3 = 2.
How do you handle a negative fractional exponent?
Work out the positive exponent and take 1 over it. 16−3/4 = 1/163/4 = 1/(⁴√16)³ = 1/2³ = 1/8 = 0.125.
Can a negative number have a fractional exponent?
Yes, when the reduced denominator is odd. (−8)1/3 = −2 and (−8)2/3 = 4. When it is even there is no real answer: (−4)1/2 is not a real number.
How do you convert a fractional exponent to radical form?
Write the denominator as the index of the root and keep the numerator as a power: xm/n = ⁿ√(xm). For example, 52/3 = ³√(5²) = ³√25.
Does it matter whether you take the root or the power first?
For a positive base, no: (ⁿ√x)m and ⁿ√(xm) are equal. Taking the root first keeps the numbers smaller: (³√8)² = 2² = 4, while ³√(8²) = ³√64 = 4.
Is a fractional exponent the same as multiplying by a fraction?
No. 82/3 is 4, while 8 × 2/3 is about 5.33. A fractional exponent is a root and a power, not a multiplication.
Is the Fractional Exponent Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Fractional Exponent Calculator to turn any fractional power into a root and a power, with the steps and exact answer shown.