Question

in the following equation, what is the value of n?
$8^{n} = 8^{-3} \cdot 8^{7}$
$n = \square$

Explanation:

Step1: Recall exponent rule

When multiplying exponents with the same base, we use the rule \(a^m \cdot a^n = a^{m + n}\). Here, the base \(a = 8\), \(m=-3\) and \(n = 7\).
So, \(8^{-3}\cdot8^{7}=8^{-3 + 7}\)

Step2: Simplify the exponent

Calculate \(-3+7 = 4\). So, \(8^{-3}\cdot8^{7}=8^{4}\)
Since \(8^{n}=8^{-3}\cdot8^{7}\), then \(8^{n}=8^{4}\). For exponential equations with the same base, the exponents must be equal. So, \(n = 4\)

Answer:

\(4\)