All subjects · Mathematics · WMA11 (P1)

Laws of indices

Index laws let you simplify powers quickly, including negative and fractional powers. You need them before you can differentiate or integrate.

The three laws

They only work when the base is the same.

Zero, negative and fractional powers

Exam habits

Take the root first, then the power: 82/3 = 22 = 4. A negative power means 'one over', not a negative number. To write an expression as a power of x, turn roots into fractional powers and move x from the bottom to a negative power: 3/sqrt(x) = 3x-1/2.

Worked example

Evaluate 16-3/4.

  1. The fourth root of 16 is 2.
  2. 23 = 8.
  3. The negative power means one over: 1/8.

Answer: 16-3/4 = 1/8

Key idea: am x an = am+n, am / an = am-n, (am)n = amn; a-n = 1/an; am/n = (nth root of a)m.

Check yourself

  1. Evaluate 272/3.
    Show answer

    9. The cube root of 27 is 3, and 32 = 9.

  2. Evaluate 16-1/2. Give your answer as a fraction in its simplest form.
    Show answer

    1/4. 161/2 = 4, and the negative power gives 1/4.

  3. Evaluate (8/27)-2/3. Give your answer as a fraction in its simplest form.
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    9/4. The negative power flips the fraction: (27/8)2/3. The cube root is 3/2, and (3/2)2 = 9/4.

  4. Write 1/(x sqrt(x)) as a single power of x.
    • x-1/2
    • x-3/2
    • x3/2
    • x-2/3
    Show answer

    x-3/2. x sqrt(x) = x1 x x1/2 = x3/2. On the bottom, it becomes x-3/2.

  5. Simplify (2x3)4.
    • 8x12
    • 2x12
    • 16x7
    • 16x12
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    16x12. Both parts are raised to the power 4: 24 = 16 and (x3)4 = x12.

  6. Given that 4x = 2x + 3, find x.
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    3. Write 4x as 22x. Then 2x = x + 3, so x = 3.

  7. Given that (x2 sqrt(x)) / x-1 = xk, find k.
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    3.5. x2 x x1/2 = x5/2. Dividing by x-1 adds 1 to the power: 5/2 + 1 = 7/2 = 3.5.

  8. Evaluate 323/5.
    Show answer

    8. The fifth root of 32 is 2, and 23 = 8.

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Last updated 7 November 2026. Spotted a mistake? Tell us.