A fractional exponent is a root in disguise. The denominator tells you which root to take and the numerator tells you which power to raise to, so x^(2/3) x^(2/3) is the cube root of x squared, ∛(x²) nthroot(x^2,3) . That one reading turns every radical you have ever met into an exponent, and every exponent rule you already trust into a rule about roots. Paste any of these into the simplify calculator as you read and watch it agree.
This is the post where roots and powers become the same subject. A radical is a fractional exponent, which means the reciprocal habit from negative exponents and the root work in radicals with variables are two views of one idea. Learn the bridge here and both sides get easier.
Why the definition has to be this
You do not have to take x^(1/2) = √x x^(1/2) on faith. It is forced by the one exponent rule this whole topic leans on: the power rule, which says a power raised to a power multiplies the exponents.
Ask what x^(1/2) x^(1/2) would have to be if the power rule keeps working. Square it, and the exponents multiply: (x^(1/2))² = x^(1/2 · 2) = x¹ = x (x^(1/2))^2 . So whatever x^(1/2) x^(1/2) is, squaring it gives x. The number whose square is x is the square root of x. There is no other choice, so x^(1/2) = √x sqrt(x) .
The same reasoning fixes every root. x^(1/3) x^(1/3) cubed is x^(1/3 · 3) = x¹ = x (x^(1/3))^3 , so it must be the cube root. The denominator n in the exponent is the root, full stop.
The numerator is the power, and it stacks on top by the same rule. Since x^(2/3) = x^(2 · 1/3) = (x^(1/3))² x^(2/3) , you take the cube root and then square it. For the full product, quotient, power, and zero rules in one place, the homepage keeps a complete rules reference. This post only needs the power rule above.
The conversion table, both directions
Because a radical and a fractional exponent say the same thing, you can trade either form for the other at any time. This is the single most useful skill on the page: rewrite a stubborn radical as an exponent so the exponent rules apply, or rewrite an exponent as a radical so you can read off the root. Here are the swaps you will use most, with the calculator input for each.
| Radical form | Exponent form | Type this |
|---|---|---|
√x sqrt(x) | x^(1/2) x^(1/2) | sqrt(x) or x^(1/2) |
∛x nthroot(x,3) | x^(1/3) x^(1/3) | nthroot(x,3) |
∛(x²) nthroot(x^2,3) | x^(2/3) x^(2/3) | nthroot(x^2,3) |
⁴√(x³) nthroot(x^3,4) | x^(3/4) x^(3/4) | nthroot(x^3,4) |
1/√x 1/sqrt(x) | x^(−1/2) x^(-1/2) | x^(-1/2) |
Read the table left to right to turn a root into an exponent, or right to left to turn an exponent into a root. The last row is the bridge to negative exponents: a minus sign in front of the fraction still means reciprocal, exactly as it did there, and the fraction underneath still means root over power.
Which comes first, the root or the power?
An expression like 16^(3/4) 16^(3/4) has two jobs in it, a fourth root and a cube. They commute, so the order never changes the answer. It changes how ugly the arithmetic gets.
Take the root first and the numbers stay small. The fourth root of 16 is 2, and then 2³ = 8 2^3 . Take the power first and you are cubing before you shrink anything: 16³ = 4096 16^3 , and then the fourth root of 4096 is still 8, but you had to know that 4096 is a fourth power to get there. Same answer, much heavier lifting.
So the rule of thumb is simple. Take the root first, then apply the power. Rooting first keeps every intermediate number as small as it can be, which is why it is easier by hand. The calculator does not care about the order, but you will.
One move comes even before that: reduce the exponent fraction. x^(4/6) x^(4/6) is the same as x^(2/3) x^(2/3) , so simplify the fraction to 2/3 2/3 and you are taking a cube root instead of a sixth root. And when the base is not a clean power, such as 7^(2/3) 7^(2/3) , leave it as an exact root rather than forcing a decimal: it is simply ∛(7²) = ∛49 nthroot(49,3) . Exact beats rounded unless the question asks for a decimal.
Negative bases and even roots
There is one honest catch, and skipping it causes real mistakes. When the denominator of the exponent is even, you are taking an even root, and even roots of negative numbers are not real.
Look at the denominator, not the sign out front. (−16)^(1/2) (-16)^(1/2) asks for the square root of a negative number, which has no real value. (−16)^(1/4) (-16)^(1/4) asks for an even root of a negative number too, so it is also not real. But an odd denominator is fine, because odd roots of negatives exist: (−8)^(1/3) = −2 (-8)^(1/3) , since (−2)³ = −8 (-2)^3 .
This is why textbooks assume x > 0 x > 0 whenever a variable sits under an even root or carries an exponent with an even denominator. The assumption is not fussiness; it is what keeps the answer real. Whenever you write x^(1/2) x^(1/2) or x^(3/4) x^(3/4) with a variable base, carry the condition x > 0 x > 0 with it.
The payoff: roots that would not combine before
Here is where the bridge earns its keep. As radicals, a square root and a cube root look like they cannot be multiplied, because their indices do not match. As exponents, they multiply in one line by adding the fractions.
Take √x · ∛x sqrt(x)*nthroot(x,3) . Rewrite each as an exponent and add the powers by the product rule: x^(1/2) · x^(1/3) = x^(1/2 + 1/3) = x^(5/6) x^(1/2)*x^(1/3) . That last form is the sixth root of x to the fifth, ⁶√(x⁵) nthroot(x^5,6) , a step that is painful with radicals and quick with exponents.
Nested roots collapse the same way. A root of a root multiplies the two fractions: √(∛x) = (x^(1/3))^(1/2) = x^(1/6) sqrt(nthroot(x,3)) , the plain sixth root. This is the habit that carries straight into radicals with variables: when a radical resists, switch it to an exponent, use the rules you already know, and switch back at the end. Because an even denominator turns up here too, keep x > 0 x > 0 in view.
Three worked examples, easy to exam-level
Watch root-over-power hold up as the problems get harder. Each step names the rule it uses.
Rational exponents, three levels
8^(1/3) - Read the fraction as root over power: the denominator 3 means cube root, and the numerator 1 means first power.
- Take the root: ∛8 = 2
nthroot(8,3), since 2³ = 82^3. - Apply the numerator power, which is 1, so the value is unchanged: 2
2.
2 16^(3/4) - Read the fraction as root over power: denominator 4 means fourth root, numerator 3 means cube.
- Take the root first (smaller numbers): ⁴√16 = 2
nthroot(16,4), since 2⁴ = 162^4. - Apply the power: 2³ = 8
2^3. Power-first gives the same answer with harder numbers: ⁴√(16³) = ⁴√4096 = 8nthroot(16^3,4).
8 x^(-3/2) - Apply the reciprocal rule for the negative exponent: x^(−3/2) = 1/x^(3/2)
1/x^(3/2). - Read the fraction as root over power: denominator 2 is a square root and numerator 3 is a cube, so x^(3/2) = √(x³) = x√x
sqrt(x^3). - Because the denominator is even (a square root), assume x > 0
x > 0so the answer is real. The result is 1/√(x³)1/sqrt(x^3), which also equals 1/(x√x)1/(x*sqrt(x)).
1/sqrt(x^3) Restriction x > 0
How do you know you are finished?
A rational-exponent problem is done when every one of these is true. Run the list before you write the final line.
- The root is fully taken. If the base is a perfect power for the denominator, the root came out to a whole number, the way ⁴√16 = 2
nthroot(16,4)did. - The power is applied and the fraction is reduced. The exponent fraction is in lowest terms, so x^(2/4)
x^(2/4)is written as x^(1/2)x^(1/2). - No negative exponent remains. A negative power means a reciprocal, so send it under the bar and keep the power positive, just as in negative exponents.
- The base condition is stated. If any denominator is even, you have written x > 0
x > 0next to the answer.
If a step is still fighting you, rewrite the whole thing in the other form. A radical that will not simplify often gives up the moment you write it as an exponent, and the reverse is just as true.
Practice: rational exponents
Practice: rational exponents
Try each one on paper first, then open the answers to check your working.
- Evaluate 25^(1/2)
25^(1/2). - Evaluate 27^(2/3)
27^(2/3). - Evaluate 32^(2/5)
32^(2/5). - Evaluate 4^(−1/2)
4^(-1/2). - Evaluate 8^(−2/3)
8^(-2/3). - Evaluate (1/9)^(3/2)
(1/9)^(3/2). - Rewrite ∛(x²)
nthroot(x^2,3)with a rational exponent. - Rewrite x^(3/4)
x^(3/4)as a radical.
Show answers
- Square root, then first power: 25^(1/2) = √25 = 5
5. - Cube root first, then square: ∛27 = 3
nthroot(27,3), then 3² = 99. - Fifth root first, then square: ⁵√32 = 2
nthroot(32,5), then 2² = 44. - Reciprocal, then square root: 4^(−1/2) = 1/√4 = 1/2
1/2. - Reciprocal of the cube root cubed… squared: ∛8 = 2
nthroot(8,3), 2² = 44, then reciprocal, 1/41/4. - Square root first, then cube: √(1/9) = 1/3
sqrt(1/9), then (1/3)³ = 1/271/27. - Denominator 3, numerator 2: ∛(x²) = x^(2/3)
x^(2/3). - Denominator 4 is the root, numerator 3 is the power: x^(3/4) = ⁴√(x³)
nthroot(x^3,4).
For the formal definition of an nth root, including the real and complex cases, Wikipedia’s Nth root page is a solid reference.
Root first, then power
Every rational exponent is a two-word instruction once you read it right: the bottom names the root, the top names the power, and taking the root first keeps the arithmetic small. Rewrite radicals as exponents when the exponent rules would help, rewrite exponents as radicals when you want to read off the root, and keep x > 0 x > 0 in view whenever an even denominator shows up. That single translation is what ties radicals with variables back to the exponents you already know.
Stuck on a fractional power? Read the denominator out loud as the root, and the problem usually solves itself.