64 raised to the power of x equal to 4² what exponential is this

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.

Learn how to solve the exponential equation 64^x = 4² using prime factorization and the properties of exponents. The solution involves rewriting 64 and 4 as powers of 2 and solving for x. Suitable for students in Grades 8-10.

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