Question

which expression is equivalent to $7^{5} \cdot 7^{-9}$?\
$\frac{1}{7^{4}}$ $\frac{1}{7^{-45}}$ $\frac{1}{7^{-4}}$ $\frac{1}{7^{45}}$

Explanation:

Step1: Apply exponent rule for multiplication

When multiplying exponents with the same base, we use the rule \(a^m \cdot a^n = a^{m + n}\). Here, \(a = 7\), \(m = 5\), and \(n=-9\). So, \(7^{5}\cdot7^{-9}=7^{5 + (-9)}\).

Step2: Simplify the exponent

Calculate \(5+(-9)=5 - 9=-4\). So, \(7^{5}\cdot7^{-9}=7^{-4}\).

Step3: Apply negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\), so \(7^{-4}=\frac{1}{7^{4}}\).

Answer:

\(\frac{1}{7^{4}}\) (the first option)