Question

what is the corresponding rule for
(a^m)^n = a^mn

a 5^-3
b b^5/b10
c (2a^4)^3

Answer 1

The rule you mentioned, \((a^m)^n = a^{mn}\), is a property of exponents known as the power of a power rule. It states that when you have a base raised to an exponent, and that entire expression is raised to another exponent, you can multiply the exponents.

Let's apply this rule to each of the expressions you've given:

a. \(5^{-3}\)

This expression involves a single base raised to a negative exponent. It doesn't involve two exponents, so the rule \( (a^m)^n = a^{mn} \) doesn't directly apply here, but you can rewrite it as:
\[
5^{-3} = \frac{1}{5^3}
\]

b. \(\frac{b^5}{b^{10}}\)

Using the quotient rule of exponents (\( \frac{a^m}{a^n} = a^{m-n} \)), we can simplify this expression:
\[
\frac{b^5}{b^{10}} = b^{5-10} = b^{-5}
\]

c. \((2a^4)^{3}\)

Using the power of a product rule \((xy)^n = x^n y^n\) and the power of a power rule, we can expand this expression:
\[
(2a^4)^3 = 2^3 \cdot (a^4)^3 = 8 \cdot a^{4 \cdot 3} = 8a^{12}
\]

So, summarizing the results:
- a: \(5^{-3}\) remains as is, or can be expressed as \(\frac{1}{5^3}\).
- b: \(\frac{b^5}{b^{10}} = b^{-5}\).
- c: \((2a^4)^3 = 8a^{12}\).