Solve $2^{5x} \div 2^x = \sqrt[5]{2^{10}}$

Step 1: Analyze the question We have an exponential equation with the same base (2) on both sides. We simplify each side using the laws of indices, then equate the powers.

Step 2: Concept (laws of indices)

• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• ⁿ√(aᵐ) = a^(m/n)

Step 3: Solution

Left side: 2^(5x) ÷ 2^x = 2^(5x − x) = 2^(4x)

Right side: ⁵√(2¹⁰) = 2^(10/5) = 2²

Equate the powers: 4x = 2 x = 2/4 = 1/2

Check: 2^(5/2) ÷ 2^(1/2) = 2^(4/2) = 2² = 4, and ⁵√(2¹⁰) = 2² = 4. ✓

Step 4: Answer x = 1/2, which is option B.

Tip: When the bases match, just "drop the base" and equate the exponents. Remember that dividing means subtracting powers, and a root means dividing the power by the root's index.