Question

question
which expression is equivalent to $5^{-2} \cdot 5^{-1}$?

answer
$\frac{1}{125}$ $\frac{1}{5}$
$\frac{1}{25}$ $25$

Explanation:

Step1: Use exponent rule \(a^m \cdot a^n = a^{m + n}\)

For \(5^{-2} \cdot 5^{-1}\), we add the exponents: \(-2 + (-1)=-3\), so it becomes \(5^{-3}\).

Step2: Use negative exponent rule \(a^{-n}=\frac{1}{a^n}\)

For \(5^{-3}\), applying the rule gives \(\frac{1}{5^3}\).

Step3: Calculate \(5^3\)

\(5^3 = 5\times5\times5 = 125\), so \(\frac{1}{5^3}=\frac{1}{125}\).

Answer:

\(\frac{1}{125}\) (the first option)