We are given the equation:

$2 \log_3{x} + 2 \log_3{x^2} = 64$

Step 1: Use logarithmic properties

We can simplify the expression using logarithmic identities.

1. Product Rule: $\log_b{(a^m)} = m \log_b{a}$.
2. Power Rule: $\log_b{(a^m)} = m \log_b{a}$.

First, let's simplify each logarithmic term:

$2 \log_3{x} + 2 \log_3{x^2}$

For the second term, use the Power Rule:

$\log_3{x^2} = 2 \log_3{x}$

So the equation becomes:

$2 \log_3{x} + 2(2 \log_3{x}) = 64$

This simplifies to:

$2 \log_3{x} + 4 \log_3{x} = 64$

$6 \log_3{x} = 64$

Step 2: Solve for $\log_3{x}$

Now divide both sides by 6:

$\log_3{x} = \frac{64}{6} = \frac{32}{3}$

Step 3: Solve for $x$

We now rewrite the equation in exponential form:

$x = 3^{\frac{32}{3}}$

Thus, the solution is:

$x = 3^{\frac{32}{3}}$

This is the exact solution. If you'd like an approximation, it can be calculated numerically.

Would you like to see the approximate decimal value, more details on logarithmic properties, or have any questions?

Here are 5 follow-up questions for further understanding:

1. How do you simplify logarithmic expressions using the product and power rules?
2. What is the difference between common logarithms and natural logarithms?
3. How can exponential equations be solved using logarithms?
4. How would you solve if the base of the logarithms was different, say 10 instead of 3?
5. Can you explain the concept of logarithms in terms of inverse functions?

Tip: Always remember the key logarithmic identities—they simplify complex expressions and make problem-solving easier!