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The image contains two parts, a chart in the upper half with examples of rational exponents and their equivalent radical forms, and an equation in the lower half for solving.

### Upper half (Converting between rational exponents and radicals):
1. \( 6^{\frac{3}{2}} \) is equivalent to \( \sqrt{6^3} \), or \( (\sqrt{6})^3 \).
2. \( 10^{\frac{3}{2}} \) is equivalent to \( \sqrt{10^3} \), or \( (\sqrt{10})^3 \).
3. \( 7^{\frac{1}{2}} \) is equivalent to \( \sqrt{7} \), or \( \sqrt{7} \).

These examples convert expressions with rational exponents into their equivalent radical forms.

### Lower half (Solving the equation):
The equation is:

\[
\left( 5^{\frac{x}{3}} \right) \left( 5^{\frac{x}{5}} \right) = 5^4
\]

To solve for \( x \), we use the property of exponents: \( a^m \times a^n = a^{m+n} \). Therefore, we can add the exponents on the left-hand side:

\[
5^{\frac{x}{3} + \frac{x}{5}} = 5^4
\]

Now, add the exponents \( \frac{x}{3} \) and \( \frac{x}{5} \). To do that, find a common denominator:

\[
\frac{x}{3} + \frac{x}{5} = \frac{5x}{15} + \frac{3x}{15} = \frac{8x}{15}
\]

This gives us:

\[
5^{\frac{8x}{15}} = 5^4
\]

Since the bases are the same (both are base 5), the exponents must be equal:

\[
\frac{8x}{15} = 4
\]

Now, solve for \( x \):

\[
8x = 4 \times 15 = 60
\]

\[
x = \frac{60}{8} = 7.5
\]

Thus, the solution is:

\[
x = 7.5
\]

---

Would you like further details on any step? Here are some related questions:

1. How do we find the common denominator when adding fractions?
2. What are the key rules for manipulating exponents?
3. How can we convert radical expressions back into rational exponents?
4. How would the equation change if the exponents had different bases?
5. What is the significance of having the same base when comparing exponents?

**Tip**: When multiplying expressions with the same base, remember that you can simply add the exponents, which simplifies complex equations.

Learn how to solve the exponent equation (5^(x/3))(5^(x/5)) = 5^4. This problem covers algebraic concepts involving the rules of exponents, including adding exponents and solving for x. Suitable for students in Grades 9-11.

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