Question

Select the equivalent expression (4³ / 5⁻2)⁵

a) 4 ¹⁵ ×5 ¹⁰
b) 4 ²⁰ ×5 ^⁵
c) 4²⁵ ×5^ ⁰
d) 4³⁰ ×5 −⁵

Answer 1

The equivalent expression for (4³ / 5⁻²)⁵ is simplified by using rules of exponents to result in 4¹⁵ × 5¹⁰, which matches option (a).

Step-by-step explanation:

The question requires us to simplify the expression (4³ / 5⁻²)⁵ and select the equivalent expression from the given options. To solve this problem, we need to apply the rules of exponents.

Here's the step-by-step breakdown:

1. Raise both numerator and denominator inside the parentheses to the power of 5. This is based on the power of a quotient rule, which states that ((a/b)^c = a^c / b^c).
2. Apply the power of a power rule where ((a^b)^c = a^(b*c)). In this case, (4³)⁵ = 4³*5 and (5⁻²)⁵ = 5⁻2*5 (this rule is analogous to our reference of underlying exponentiated quantities).
3. Simplify the expression by calculating the new exponents: 4³*5 = 4¹⁵ and 5⁻²*5 = 5⁻¹⁰.
4. According to multiplication rule of exponents, we can turn division of powers with the same base into subtraction of exponents (when we have (a^b / a^c = a^(b-c))), but since we have multiplied the exponents, in this case, we combine the two components into a single product.

Therefore, the simplified form of the given expression is 4⁹⁵ × 5¹⁰, which matches option (a).