Question

what is the value of k in the equation \\(\frac{1}{3^7} = 3^k\\)? \\(k = \square\\)

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that $\frac{1}{a^n} = a^{-n}$ for any non - zero real number $a$ and integer $n$. Applying this rule to $\frac{1}{3^{7}}$, we get $\frac{1}{3^{7}}=3^{-7}$.

Step2: Equate the exponents

We have the equation $\frac{1}{3^{7}} = 3^{k}$, and from Step 1 we know that $\frac{1}{3^{7}}=3^{-7}$. So, by the property of exponential functions (if $a^m=a^n$, then $m = n$ for $a>0,a
eq1$), we can say that since $3^{-7}=3^{k}$, then $k=-7$.

Answer:

$-7$