Question

Evaluate:

(5³)⁴ × (5^–2)⁵

Answer 1

Evaluation of the Expression

The given expression is (5³)⁴ × (5^–2)⁵.

Step-by-Step Solution

Apply the power of a power rule:.

When a power is raised to another power, the exponents are multiplied.

This rule is expressed as (a^m)ⁿ = a^{m × n}.

For the first term, (5³)⁴, the exponents are multiplied: 3 × 14 = 12. Thus, (5³)⁴ = 5¹².

For the second term, (5^–2)⁵, the exponents are multiplied: –2 × 5 = –10. Thus, (5^–2)⁵ = 5^–10.

Rewrite the expression:

The expression is now 5¹² × 5^–10.

Apply the product rule for exponents:

When multiplying powers with the same base, the exponents are added.

This rule is expressed as a^m × aⁿ = a^{m + n}.

For 5¹² × 5^–10, the exponents are added: 12 + (–10) = 12 – 10 = 2.

Calculate the final value:

The expression simplifies to 5².

5² = 5 × 5 = 25.