To solve the equation:

$64^x = 4^2$

we'll break down each side of the equation in terms of prime factors and solve for $x$.

Step 1: Rewrite the terms in terms of base 2

Notice that both 64 and 4 can be rewritten as powers of 2:

1. $64 = 2^6$, so $64^x = (2^6)^x = 2^{6x}$.
2. $4 = 2^2$, so $4^2 = (2^2)^2 = 2^4$.

This gives us the equation:

$2^{6x} = 2^4$

Step 2: Set exponents equal to each other

Since the bases are the same (both are powers of 2), we can set the exponents equal to each other:

$6x = 4$

Step 3: Solve for $x$

Divide both sides by 6:

$x = \frac{4}{6} = \frac{2}{3}$

Final Answer

The solution is:

$x = \frac{2}{3}$

Would you like further details on this, or have any other questions?

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Here are five related questions you might find interesting:

1. How would you solve $64^x = 8$?
2. Can you rewrite $81$ as a power of a different base and solve equations involving it?
3. What if the equation was $4^x = 64$? How would you solve for $x$?
4. How do we solve exponential equations where the bases are not powers of each other?
5. Can logarithms help solve more complex exponential equations?

Tip: When solving exponential equations, try to rewrite each side with the same base. This simplifies the equation by allowing you to set the exponents equal to each other.