Question

question
which expression is equivalent to \\(\frac{5^{-1}}{5^{-2}} cdot 5^{-3}\\)?

answer
\\(\bigcirc\\) \\(5^4\\) \\(\bigcirc\\) \\(5^{-2}\\)
\\(\bigcirc\\) \\(5^{-4}\\) \\(\bigcirc\\) 1

Explanation:

Step1: Simplify the fraction using exponent rule

Recall the exponent rule \(\frac{a^m}{a^n}=a^{m - n}\). For \(\frac{5^{-1}}{5^{-2}}\), we have \(m=-1\) and \(n = - 2\), so \(\frac{5^{-1}}{5^{-2}}=5^{-1-(-2)}=5^{-1 + 2}=5^{1}\).

Step2: Multiply with the other term using exponent rule

Recall the exponent rule \(a^m\cdot a^n=a^{m + n}\). Now we multiply \(5^{1}\) with \(5^{-3}\). Here \(m = 1\) and \(n=-3\), so \(5^{1}\cdot5^{-3}=5^{1+(-3)}=5^{-2}\).

Answer:

\(5^{-2}\) (corresponding to the option with \(5^{-2}\))