Matematica

Laws of Exponents: Solved Exercises Step by Step

The laws of exponents explained simply, with the rules for multiplying, dividing and raising powers, plus five solved exercises of increasing difficulty with verified answers.

Recommended for: Grade 8 · Grade 9

The laws of exponents let you simplify products, quotients and powers of powers without doing every calculation. The basic rule is: when the base is the same, to multiply you add the exponents (aᵐ · aⁿ = aᵐ⁺ⁿ) and to divide you subtract them (aᵐ ÷ aⁿ = aᵐ⁻ⁿ).

The core rules

All the properties apply only when the base is the same:

  • Product: aᵐ · aⁿ = aᵐ⁺ⁿ (add the exponents);
  • Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the exponents);
  • Power of a power: (aᵐ)ⁿ = aᵐ·ⁿ (multiply the exponents).

Two more properties are useful when the base changes but the exponent stays the same: (a · b)ⁿ = aⁿ · bⁿ and (a ÷ b)ⁿ = aⁿ ÷ bⁿ.

Zero exponent and negative exponent

Two cases worth memorising because they come up often: any nonzero number raised to 0 gives 1 (a⁰ = 1), while a negative exponent means the reciprocal (a⁻ⁿ = 1/aⁿ). Both results follow directly from the quotient rule.

How to tackle the exercises

The method is always the same: check whether the bases are equal, apply the right property by working on the exponents, and only at the end compute the numerical value. In the exercises below you will find the same approach at increasing difficulty, from simple products up to an expression with negative exponents to handle carefully.

Solved exercises

1. Simplify: 2³ · 2⁴ base

Show solution
  1. The bases are equal (2), so in a product you add the exponents.
  2. 2³ · 2⁴ = 2^(3+4) = 2⁷.
  3. 2⁷ = 128.

Answer: 2⁷ = 128

2. Simplify: 5⁶ ÷ 5⁴ base

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  1. The bases are equal (5), so in a quotient you subtract the exponents.
  2. 5⁶ ÷ 5⁴ = 5^(6−4) = 5².
  3. 5² = 25.

Answer: 5² = 25

3. Simplify: (3²)³ intermedio

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  1. This is a power of a power: multiply the exponents.
  2. (3²)³ = 3^(2·3) = 3⁶.
  3. 3⁶ = 729.

Answer: 3⁶ = 729

4. Simplify: (10³ · 10⁻¹) ÷ 10² intermedio

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  1. In the product, add the exponents: 10³ · 10⁻¹ = 10^(3+(−1)) = 10².
  2. In the quotient, subtract: 10² ÷ 10² = 10^(2−2) = 10⁰.
  3. Any nonzero number raised to 0 equals 1.

Answer: 10⁰ = 1

5. Simplify: (x⁴ · x⁻⁶)⁻² avanzato

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  1. Inside the brackets, add the exponents: x⁴ · x⁻⁶ = x^(4+(−6)) = x⁻².
  2. Now it is a power of a power: (x⁻²)⁻² = x^((−2)·(−2)) = x⁴.
  3. The two negative signs give a positive exponent.

Answer: x⁴

FAQ

What is a power with exponent 0?

Any nonzero number raised to the power 0 equals 1: for example 7⁰ = 1. The expression 0⁰, however, is undefined.

What does a negative exponent mean?

A negative exponent means the reciprocal: a⁻ⁿ = 1/aⁿ. For example 2⁻³ = 1/2³ = 1/8.

Can I add the exponents even when the bases are different?

No. The product rule only works when the base is the same: 2³ · 3² cannot be simplified by adding exponents, because the bases 2 and 3 are different.