Question

solve. write your answer as an integer or a fraction in simplest form.\\(64^{2x} = 8\\)\\(x = \square\\)

Explanation:

Step1: Express bases as powers of 2

We know that \(64 = 2^6\) and \(8 = 2^3\). Substitute these into the equation:
\((2^6)^{2x}=2^3\)

Step2: Simplify left side using exponent rule

Using the exponent rule \((a^m)^n = a^{mn}\), we get:
\(2^{12x}=2^3\)

Step3: Set exponents equal (since bases are same)

If \(a^m = a^n\), then \(m = n\). So:
\(12x = 3\)

Step4: Solve for x

Divide both sides by 12:
\(x=\frac{3}{12}=\frac{1}{4}\)

Answer:

\(\frac{1}{4}\)