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Fractional exponents: what xm/n means and how to simplify

A fraction in the exponent looks like a new kind of math, but it's two things you already know stacked together: a root and a power. The bottom of the fraction is the root, the top is the power.

Short answer

x1/n is the nth root of x: 91/2 = √9 = 3 and 81/3 = 3√8 = 2. For xm/n, the denominator is the root and the numerator is the power: 82/3 = (3√8)2 = 22 = 4. Take the root first and the numbers stay small. A negative fractional exponent still means "one over": 8−2/3 = 1/4, a positive number.

Why it works

The exponent rule (xa)b = xab is what forces the meaning. Square 91/2:

  1. (91/2)2 = 9(1/2)·2Power of a power: multiply the exponents.
  2. 9(1/2)·2 = 91 = 9Half of 2 is 1.
  3. 91/2 = 3So 91/2 is a number whose square is 9. That's the definition of √9, and the positive choice is 3.

The same argument with cubes shows x1/3 is the cube root, and in general x1/n = n√x. Then any fraction splits into "one over n" times m:

xm/n = (x1/n)m = (n√x)m, and also xm/n = (xm)1/n = n√xm.

Both orders give the same answer for a positive base. The root-first order is the one to use by hand, because it shrinks the number before the power grows it.

Example 1: root first versus power first

Evaluate 272/3.

  1. 272/3 = (3√27)2The denominator 3 is a cube root. The numerator 2 is a square.
  2. = 323³ = 27, so the cube root of 27 is 3.
  3. = 9Small numbers the whole way.

Power first works too, but look at the size of the middle step: 272 = 729, and then you need the cube root of 729, which is 9. Same answer, much harder to do without a calculator.

Example 2: a negative fractional exponent

Evaluate 16−3/4.

  1. 16−3/4 = 1163/4The minus sign means "one over." It doesn't make anything negative. (See negative and zero exponents.)
  2. = 1(4√16)3Denominator 4 is a fourth root, numerator 3 is a cube.
  3. = 1232⁴ = 16, so the fourth root of 16 is 2.
  4. = 18Positive, and smaller than 1, which is what a negative exponent on a number bigger than 1 always gives.

A fraction as the base flips first: (4/9)−3/2 = (9/4)3/2 = (3/2)3 = 27/8.

Example 3: switching between radical and exponent form

Exponent form is easier to simplify, because all the exponent rules apply. Write √x · 3√x as a single radical, for x > 0.

  1. x1/2 · x1/3Each root becomes a fractional exponent: square root is 1/2, cube root is 1/3.
  2. = x1/2 + 1/3Same base, multiplying, so add the exponents. This step is impossible to do in radical form.
  3. = x5/6Common denominator: 3/6 + 2/6 = 5/6.
  4. = 6√x5Back to radical form: denominator 6 is the root, numerator 5 is the power.

Example 4: a coefficient and a variable

Simplify (8x6)2/3.

  1. 82/3 · (x6)2/3The exponent outside the parentheses applies to every factor inside, including the 8.
  2. = (3√8)2 · x6 · 2/3Root first on the number. Power of a power on the variable: multiply exponents.
  3. = 4x42² = 4, and 6 · 2/3 = 4.

Negative bases: odd roots are fine, even roots are not

(−8)1/3 = −2, because (−2)3 = −8. A negative number cubed stays negative, so every negative number has a real cube root (and fifth root, and so on).

(−16)1/4 is not a real number. Any real number to the fourth power is zero or positive, so nothing raised to the fourth gives −16. The same goes for (−9)1/2 = √−9. (Those are where complex numbers come in.) Some calculators also refuse (−8)^(1/3) even though the answer is a perfectly good −2, so if yours does, take the cube root of 8 and put the sign back.

Common mistakes, and the exact line they happen on

1. Multiplying by the fraction

  1. 82/3 = 8 · 23 = 163This is the mistake. An exponent is never a multiplier, whether it's 2 or 2/3. 8² isn't 8 · 2 either. 82/3 = (∛8)² = 2² = 4.

2. Putting the root and the power in the wrong places

  1. 82/3Evaluate.
  2. = (√8)3 = 16√2 ≈ 22.6This is the mistake. That's a square root cubed, which is 83/2. The bottom of the fraction is the root: 82/3 is a cube root squared, which is 4.

A memory hook: the root is the denominator because a root sits "down at the base," like the roots of a plant.

3. Making the answer negative

  1. 27−1/3 = −3This is the mistake. The minus in the exponent means "one over," not "negative." 27−1/3 = 1/∛27 = 1/3, a positive number.

4. Adding two roots by adding the exponents

  1. x1/2 + x1/2 = x1 = xThis is the mistake. Exponents add when you multiply powers of the same base, not when you add them. These are like terms: x1/2 + x1/2 = 2x1/2. At x = 9: √9 + √9 = 6, but the wrong answer says 9.

What is true is x1/2 · x1/2 = x1 = x: a square root times itself gives back the number.

5. Taking an even root of a negative number

  1. (−16)1/2 = −4This is the mistake. Check it by squaring: (−4)² = 16, not −16. No real number squares to a negative, so (−16)1/2 has no real value. The odd root (−8)1/3 = −2 does work, because (−2)³ = −8.

Every mistake on this page fails the same check: raise your answer to the power the root undoes. If 82/3 were 16/3, then cubing it would have to give 82 = 64, and (16/3)3 is about 152.

Practice

  1. Evaluate 323/5.

    Show answer

    Fifth root first: 5√32 = 2, then cube: 23 = 8.

  2. Evaluate 81−3/4.

    Show answer

    1/813/4 = 1/(4√81)3 = 1/33 = 1/27.

  3. Write 3√x2 in exponent form and y3/4 in radical form.

    Show answer

    3√x2 = x2/3. y3/4 = 4√y3, which can also be written (4√y)3.

  4. Evaluate (−27)2/3.

    Show answer

    The cube root of −27 is −3, because (−3)3 = −27. Then (−3)2 = 9.

  5. Simplify (16x8)3/4.

    Show answer

    163/4 = (4√16)3 = 23 = 8, and (x8)3/4 = x8 · 3/4 = x6. Answer: 8x6.

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