Every exponent law with its own worked example — multiplying, dividing, powers of powers and products, zero and negative exponents — plus combined exam-style questions and a full worksheet with answers.
Exponents (indices) are shorthand for repeated multiplication: x4 means x × x × x × x. Every 'law' below is just that definition applied carefully — so if you forget a rule, you can rebuild it in seconds. This guide covers every exponent law and every combination an exam can ask.
In x4, the base is x and the exponent is 4. The laws only work when the bases are the same.
1Law 1: Multiplying (same base): ADD the exponents
- 1Multiply the numbers: 2 × 5 = 10.Coefficients multiply normally.
- 2Add the exponents of x: x3 · x4 = x7.Same base, so add: 3 + 4 = 7.
- 3Answer: 10x7.Combine the number and the variable part.
2Law 2: Dividing (same base): SUBTRACT the exponents
- 1Divide the numbers: 12 ÷ 3 = 4.Coefficients divide normally.
- 2Subtract the exponents of x: x5 ÷ x2 = x3.Same base, so subtract: 5 − 2 = 3.
- 3Answer: 4x3.Combine the parts.
3Law 3: Power of a power: MULTIPLY the exponents
- 1Multiply the exponents: 3 × 4 = 12.A power raised to a power multiplies.
- 2Answer: x12.The base stays x.
4Law 4: Power of a product: give EVERY factor the power
- 1Apply the power 3 to every factor inside: 2 and x2.Both the number and the variable are raised to 3.
- 223 = 8 and (x2)3 = x6.Cube the 2; multiply exponents for the x.
- 3Answer: 8x6.The coefficient 2 becomes 8 — don't forget it.
5Law 5: Zero exponent: the answer is 1
Anything (except 0) to the power 0 equals 1. Why? Because x3 ÷ x3 = x0, but any number divided by itself is 1.
- 1In 3x0, only the x has the power 0, so x0 = 1 → 3 × 1 = 3.The power sits on x only, not the 3.
- 2In (3x)0, the whole bracket is to the power 0 → 1.Brackets mean everything inside gets the power.
6Law 6: Negative exponents: 'one over'
A negative power means take the reciprocal — it does not make the answer negative.
- 1The x−3 moves to the denominator as x3.Negative exponent means 'one over'.
- 2The 2 stays on top (its exponent is +1).Only the x has the negative power.
- 3Answer: 2 ⁄ x3.= 2⁄x3, a positive-exponent form.
7Putting it together — combined exam questions
Real exam questions mix several laws. Work from the inside out: brackets first, then multiply/divide.
- 1Bracket first (Law 4): (2x2)3 = 23x6 = 8x6.Power of a product — cube the 2 as well.
- 2Multiply by x−1 (Law 1): 8x6 × x−1 = 8x6−1.Same base, add exponents: 6 + (−1) = 5.
- 3Answer: 8x5.A single positive-exponent term.
- 1Numbers: 6 ÷ 2 = 3.Divide the coefficients.
- 2x: x4 ÷ x2 = x2. y: y2 ÷ y5 = y−3.Subtract exponents for each base separately.
- 3Write with positive exponents: 3x2 ⁄ y3.y−3 becomes 1⁄y3, so it drops to the denominator.
8Scientific notation (Grade 8–9)
Scientific notation writes a number as a value between 1 and 10 times a power of 10 — useful for very large or very small numbers.
- 1Place the decimal after the first non-zero digit: 4,5.The value must be between 1 and 10.
- 2Count how many places the decimal moved: 4 places → ×104.45 000 → 4,5 needs 4 jumps.
- 3Answer: 4,5 × 104.Large number → positive power of 10.
Two classic slips: (1) forgetting the coefficient gets the power — (2x)3 is 8x3, not 2x3; (2) thinking a negative exponent makes the answer negative — it makes a fraction, not a minus.
Numbers and variables are handled separately. Deal with the coefficients (multiply/divide the numbers) and the exponents (add/subtract/multiply) as two separate jobs, then recombine.
Every law above is examinable in Grade 8–11 Paper 1. Drill the worksheet until you can spot which law applies at a glance — exponents are among the fastest, most reliable marks in the paper.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Simplify x4 · x3 (Law 1)
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Add exponents → x7 - 2Simplify 3a2 · 4a5 (Law 1)
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3×4 = 12, add exponents → 12a7 - 3Simplify m8 ÷ m3 (Law 2)
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Subtract exponents → m5 - 4Simplify 20x6 ⁄ 5x2 (Law 2)
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20÷5 = 4, subtract exponents → 4x4 - 5Simplify (a3)2 (Law 3)
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Multiply exponents → a6 - 6Simplify (2y4)3 (Law 4)
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23 = 8, ×4×3 → 8y12 - 7Simplify (3x2)2 (Law 4)
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32 = 9 → 9x4 - 8Evaluate 7x0 (Law 5)
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x0 = 1 → 7 - 9Evaluate (5y)0 (Law 5)
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Whole bracket to power 0 → 1 - 10Write 4x−2 with a positive exponent (Law 6)
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4 ⁄ x2 - 11Simplify (2x3)2 · x (combined)
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(2x3)2 = 4x6, ×x → 4x7 - 12Simplify (12a5b2) ⁄ (4a2b6) (combined)
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12÷4 = 3, a5−2 = a3, b2−6 = b−4 → 3a3 ⁄ b4 - 13Write 3 200 000 in scientific notation
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3,2 × 106 - 14Write 0,0007 in scientific notation
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Small number → negative power → 7 × 10−4
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