Power Play

512, 2^9, 64 × 8, (2^3)^3

8^3 = 8 × 8 × 8 = 512. Also, 8 = 2^3, so 8^3 = (2^3)^3 = 2^9. And 8^3 = 8^2 × 8 = 64 × 8. The value 24 is incorrect (that would be 8 × 3).

5^4

Using the rule a^m ÷ a^n = a^(m-n): 5^7 ÷ 5^3 = 5^(7-3) = 5^4. When dividing powers with the same base, subtract the exponents. Common mistake: adding exponents instead of subtracting.

1/8

2^(-3) = 1/2^3 = 1/8. Negative exponent rule: a^(-n) = 1/a^n. So 2^(-3) = 1/2^3 = 1/8. Common mistake: making the answer negative.

2.5 × 10^(-5)

0.000025 = 2.5 × 10^(-5). Move decimal point 5 places right to get 2.5, so exponent is -5. Scientific notation: coefficient between 1-10 multiplied by power of 10.