Question

which expression is equivalent to $2^{-5} \cdot 10^{-5}$?
\\(\frac{1}{20^{-5}}\\) \\(20^{25}\\)
\\(20^{-10}\\) \\(20^{-5}\\)

Explanation:

Step1: Recall exponent rule for product

When multiplying two numbers with the same exponent, we can use the rule \(a^n \cdot b^n=(ab)^n\). Here, \(2^{-5}\cdot10^{-5}\) has the same exponent \(-5\), so we can apply this rule.
\(2^{-5}\cdot10^{-5}=(2\times10)^{-5}\)

Step2: Calculate the product inside the parentheses

Calculate \(2\times10 = 20\), so the expression becomes \(20^{-5}\).
We can also check the first option: \(\frac{1}{20^{-5}}=20^{5}\) (by the rule \(a^{-n}=\frac{1}{a^n}\), so \(\frac{1}{a^{-n}} = a^n\)), which is not equal. The second option \(20^{25}\) is incorrect. The third option \(20^{-10}\) would be if we added exponents, but we have the same exponent for multiplication, so we multiply the bases. So the correct equivalent expression is \(20^{-5}\).

Answer:

\(20^{-5}\) (the option with \(20^{-5}\))