Question

solve. write your answer as an integer or a fraction in simplest form.
$4^{2x} = 64$
$x = \boxed{\frac{\square}{\square}}$

Explanation:

Step1: Express 4 and 64 as powers of 2

We know that \(4 = 2^2\) and \(64 = 2^6\). So we can rewrite the equation \(4^{2x}=64\) as \((2^2)^{2x}=2^6\).

Step2: Simplify the left side using exponent rules

Using the exponent rule \((a^m)^n=a^{mn}\), the left side becomes \(2^{4x}\). So our equation is now \(2^{4x}=2^6\).

Step3: Set the exponents equal

Since the bases are the same and the equation holds, we can set the exponents equal to each other: \(4x = 6\).

Step4: Solve for x

Divide both sides of the equation \(4x = 6\) by 4: \(x=\frac{6}{4}\). Simplify the fraction by dividing numerator and denominator by 2: \(x=\frac{3}{2}\).

Answer:

\(\frac{3}{2}\)