https://file.aiquickdraw.com/mathbot/file/17289663694272ikc8dd5.jpg

The equation in the image is:

\[
\left( x + 9 \right)^{\frac{3}{5}} + 7 = 15
\]

Let's solve it step-by-step:

### Step 1: Isolate the term with the exponent
First, subtract 7 from both sides of the equation to get rid of the constant term:

\[
\left( x + 9 \right)^{\frac{3}{5}} = 15 - 7
\]
\[
\left( x + 9 \right)^{\frac{3}{5}} = 8
\]

### Step 2: Raise both sides to the reciprocal of the exponent
To eliminate the fractional exponent \( \frac{3}{5} \), raise both sides to the power of \( \frac{5}{3} \):

\[
\left( \left( x + 9 \right)^{\frac{3}{5}} \right)^{\frac{5}{3}} = 8^{\frac{5}{3}}
\]
\[
x + 9 = 8^{\frac{5}{3}}
\]

Now, calculate \( 8^{\frac{5}{3}} \). Since \( 8 = 2^3 \), we can simplify this:

\[
8^{\frac{5}{3}} = \left( 2^3 \right)^{\frac{5}{3}} = 2^5 = 32
\]

Thus:

\[
x + 9 = 32
\]

### Step 3: Solve for \( x \)
Subtract 9 from both sides:

\[
x = 32 - 9
\]
\[
x = 23
\]

### Final Solution:
\[
x = 23
\]

Would you like a detailed breakdown of any of these steps, or do you have any questions?

---

Here are 5 related questions to extend your understanding:
1. How would you solve the equation if the exponent were \( \frac{2}{3} \) instead of \( \frac{3}{5} \)?
2. What happens if the base inside the parentheses is negative, e.g., \( (x - 9)^{\frac{3}{5}} \)?
3. Can you explain the process of raising a number to a fractional power?
4. How would the solution change if \( 15 \) on the right-hand side were a different number?
5. What if the equation were \( \left( x + 9 \right)^{\frac{5}{3}} + 7 = 15 \)? How would you solve it?

**Tip:** When solving equations involving fractional exponents, always remember to check if the base is non-negative, since negative numbers raised to a fractional power can produce complex numbers.

Learn how to solve the equation (x + 9)^(3/5) + 7 = 15 step-by-step. This problem covers algebraic concepts such as solving equations with fractional exponents and isolating terms. Suitable for high school students in Grades 9-12.

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