Problem
Simplify: (2³ × 2⁴) ÷ 2² × (2²)³
📝Step-by-Step Solution
1
Apply the Product Rule first
When multiplying same bases, add exponents: 2³ × 2⁴ = 2³⁺⁴ = 2⁷
2³ × 2⁴ = 2⁷
2
Apply the Power Rule
When raising a power to a power, multiply exponents: (2²)³ = 2²ˣ³ = 2⁶
(2²)³ = 2⁶
3
Rewrite the expression
Now we have: 2⁷ ÷ 2² × 2⁶
2⁷ ÷ 2² × 2⁶
4
Apply the Quotient Rule
When dividing same bases, subtract exponents: 2⁷ ÷ 2² = 2⁷⁻² = 2⁵
2⁷ ÷ 2² = 2⁵
5
Apply the Product Rule again
Multiply: 2⁵ × 2⁶ = 2⁵⁺⁶ = 2¹¹
2⁵ × 2⁶ = 2¹¹
6
Calculate (optional)
2¹¹ = 2048
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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