Question

what is the value of m in the equation \\(\frac{1}{6^4} = 6^m\\)?
\\(m = \square\\)
if a is any nonzero value and n is an integer, then \\(\frac{1}{a^n} = a^{-n}\\).
an error has occurred. please enter a number.

Explanation:

Step1: Apply the negative exponent rule

Given the rule \(\frac{1}{a^n}=a^{-n}\), for \(\frac{1}{6^4}\), we can rewrite it using this rule. Here, \(a = 6\) and \(n = 4\), so \(\frac{1}{6^4}=6^{-4}\).

Step2: Compare with the given equation

The equation is \(\frac{1}{6^4}=6^m\). Since we have \(\frac{1}{6^4}=6^{-4}\), by the transitive property of equality (if \(A = B\) and \(B = C\), then \(A = C\)), we can say that \(6^{-4}=6^m\).
For exponential functions with the same base (\(a^x=a^y\) implies \(x = y\) when \(a>0,a
eq1\)), here the base is \(6\) (which is greater than \(0\) and not equal to \(1\)), so we can equate the exponents. Thus, \(m=-4\).

Answer:

\(-4\)