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Grade 10 · Exponents · 11 min read

Laws of Exponents & Rational Exponents (Grade 10)

The exponent laws extended to negative and fractional powers, converting between roots and rational exponents, simplifying harder expressions, and solving exponential equations by matching bases.

Exponents are the single most examined skill in Grade 10 Term 1. The laws themselves are unchanged from Grade 9 — what is new is that the powers may now be negative or fractional, and the same laws still hold.

1The laws (unchanged)

xm × xn = xm+n  ·  xm ÷ xn = xm−n
(xm)n = xmn  ·  xmym = (xy)m
x0 = 1  ·  x−n = 1xn  (x ≠ 0)
⚠️

A negative exponent never makes the answer negative. 3−2 = 19, not −9.

2Rational (fractional) exponents

x½=√xsquare rootx⅓=∛xcube rootxm/n=(ⁿ√x)mroot n, power mdenominator = root, numerator = power
In a fractional exponent the denominator gives the root and the numerator gives the power.
x1/n = nx   ·   xm/n = (nx)m
Worked ExampleCalculate 272/3 without a calculator
  1. 1
    The denominator 3 says take the CUBE root: 27 = 3.
    Always do the root first — the numbers stay small.
  2. 2
    The numerator 2 says square it: 3² = 9.
  3. 3
    272/3 = 9.
    Doing the power first (27² = 729) works but is far harder.
Worked ExampleCalculate 16−3/4
  1. 1
    The negative exponent means take the reciprocal: 1163/4.
    Deal with the minus sign first.
  2. 2
    163/4: fourth root of 16 is 2, then 2³ = 8.
    Root, then power.
  3. 3
    Answer: 18.
💡

Order of attack: negative sign → root → power. Taking the root before the power keeps every number small enough to do in your head.

3Simplifying expressions

Worked ExampleSimplify: 12x5y34x2y7
  1. 1
    Numbers: 12 ÷ 4 = 3.
  2. 2
    x: 5 − 2 = 3, giving x³.
    Subtract exponents when dividing.
  3. 3
    y: 3 − 7 = −4, giving y−4.
  4. 4
    = 3x3y4
    Final answers are normally written with positive exponents.
Worked ExampleSimplify: 2x+32x
  1. 1
    Same base, so subtract the exponents: (x + 3) − x.
  2. 2
    = 2³ = 8.
    The x disappears entirely — a favourite exam trick.

4Exponential equations

If two powers with the same base are equal, their exponents must be equal.

If am = an then m = n
Worked ExampleSolve: 2x = 32
  1. 1
    Write 32 as a power of 2: 32 = 25.
    Make the bases match.
  2. 2
    2x = 25, so x = 5.
Worked ExampleSolve: 32x−1 = 81
  1. 1
    81 = 34.
    Same base.
  2. 2
    2x − 1 = 4.
    Equate the exponents.
  3. 3
    2x = 5, so x = 2,5.
⚠️

You may only equate exponents once the bases are identical. From 2x = 3x you cannot say x = x — rewrite one side first.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet✅ Memo📝 Exam paper🗝️ Marking guide
  1. 1
    Simplify: x7 × x−3
    Show answer ▾
    x4
  2. 2
    Simplify: (2x3)4
    Show answer ▾
    16x12
  3. 3
    Calculate: 5−2
    Show answer ▾
    125
  4. 4
    Calculate: 91/2
    Show answer ▾
    3
  5. 5
    Calculate: 82/3
    Show answer ▾
    (8)² = 2² = 4
  6. 6
    Calculate: 272/3
    Show answer ▾
    (27)² = 3² = 9
  7. 7
    Calculate: 163/4
    Show answer ▾
    (∜16)³ = 2³ = 8
  8. 8
    Calculate: 16−3/4
    Show answer ▾
    18
  9. 9
    Simplify: 2x+32x
    Show answer ▾
    2³ = 8
  10. 10
    Solve: 2x = 64
    Show answer ▾
    64 = 26 → x = 6
  11. 11
    Solve: 3x+1 = 27
    Show answer ▾
    x + 1 = 3 → x = 2
  12. 12
    Solve: 52x = 125
    Show answer ▾
    2x = 3 → x = 1,5
🧠

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