Medium📚 Exponents

Exponent Rules Example

Grade 8Key Stage 3Årskurs 8-9

Problem

Simplify: (2³ × 2⁴) ÷ 2² × (2²)³

📝Step-by-Step Solution

Apply the Product Rule first

When multiplying same bases, add exponents: 2³ × 2⁴ = 2³⁺⁴ = 2⁷

2³ × 2⁴ = 2⁷

Apply the Power Rule

When raising a power to a power, multiply exponents: (2²)³ = 2²ˣ³ = 2⁶

(2²)³ = 2⁶

Rewrite the expression

Now we have: 2⁷ ÷ 2² × 2⁶

2⁷ ÷ 2² × 2⁶

Apply the Quotient Rule

When dividing same bases, subtract exponents: 2⁷ ÷ 2² = 2⁷⁻² = 2⁵

2⁷ ÷ 2² = 2⁵

Apply the Product Rule again

Multiply: 2⁵ × 2⁶ = 2⁵⁺⁶ = 2¹¹

2⁵ × 2⁶ = 2¹¹

Calculate (optional)

2¹¹ = 2048

✅Final Answer

2¹¹ = 2048

The three key exponent rules: (1) Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents), (2) Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract exponents), (3) Power Rule: (aᵐ)ⁿ = aᵐˣⁿ (multiply exponents). These only work when the bases are the same!

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