Rational exponent is one of 1 radicals formula in the algebra section of this library, and it is used at high school level.
Why rational exponent works
Fractional exponents are defined so that the multiplying rule keeps working. Raising a to the one-over-n to the power n multiplies the exponents to 1, returning a itself, and that is precisely what an nth root is supposed to do. The numerator then just applies the ordinary power on top.
What each symbol means
$n$ is the root index and $m$ is the power.
Rational exponent: when it holds
Over the reals, if $n$ is even then $a\ge0$; $n\ne0$.
When it stops applying
With a negative base and an even root there is no real value, since no real number squares to a negative. Worse, the fraction cannot be freely reduced: reading (-8) to the two-sixths as the sixth root of 64 gives 2, while reducing to one-third first and then squaring gives 4.
Rational exponent: a worked example
$27^{2/3}=(\sqrt[3]{27})^2=9$.
The mistake to avoid
What people do: Reading 27 to the two-thirds as 27 squared divided by 3.
Why it goes wrong: That gives 729 over 3, which is 243, while the true value is 9. The denominator of the exponent is a root instruction, not a division instruction.
Do this instead: Split the exponent in two: take the cube root of 27 to get 3, then square it to get 9. Doing the root first keeps the numbers small.
Rational exponent: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is the root index and $m$ is the power.
- Check the conditions before substituting. Over the reals, if $n$ is even then $a\ge0$; $n\ne0$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Radicals
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where rational exponent comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about rational exponent
Should I take the root first or the power first?
For a positive base either order gives the same answer, but the root first keeps the arithmetic small. Cube-rooting 27 to 3 and squaring is easier than squaring to 729 and then cube-rooting.
What does a negative fractional exponent mean?
It combines a reciprocal with a root and a power. Since 8 to the two-thirds is 4, the value of 8 to the negative two-thirds is 1 over 4, which is 0.25.
Is the square root symbol the same as an exponent of one half?
Yes, they are two notations for the same operation. Writing it as an exponent is often more convenient because the exponent rules then apply without any special cases.
How do I enter this on a calculator?
Wrap the whole fraction in brackets. Typing 27^2/3 makes the calculator divide by 3 at the end, so you must type 27^(2/3) to get 9.