The image shows the following equation:

$\left( \frac{4}{3} \right)^{8x} \times \left( \frac{4}{3} \right)^2 = \left( \frac{4}{3} \right)^5$

We are tasked with finding the value of $x$.

Solution:

We can combine the terms on the left-hand side of the equation using the property of exponents:

$a^m \times a^n = a^{m+n}$

So, we have:

$\left( \frac{4}{3} \right)^{8x + 2} = \left( \frac{4}{3} \right)^5$

Now, since the bases are the same, we can equate the exponents:

$8x + 2 = 5$

Now, solve for $x$:

$8x = 5 - 2$

$8x = 3$

$x = \frac{3}{8}$

Thus, the value of $x$ is $\frac{3}{8}$.

Would you like more details on this solution or have any questions?

Related Questions:

1. How do you solve equations with exponents that have different bases?
2. What are the properties of exponents, and how can they be applied to simplify equations?
3. How do you solve logarithmic equations involving exponentials?
4. Can you explain how to solve fractional exponents?
5. How would the solution change if the exponents were negative?

Tip: Always remember to use the properties of exponents when the base of the exponential terms is the same. It makes solving equations much simpler!