We are given the equation:

\[
(x + 5)^{\frac{3}{2}} = 8
\]

To solve this, follow these steps:

### Step 1: Isolate the expression with the exponent.
We start by eliminating the exponent \(\frac{3}{2}\) from the equation. We can do this by raising both sides of the equation to the power of \(\frac{2}{3}\), which is the reciprocal of \(\frac{3}{2}\):

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

This simplifies to:

\[
x + 5 = 8^{\frac{2}{3}}
\]

### Step 2: Evaluate \(8^{\frac{2}{3}}\).
We know that:

\[
8^{\frac{2}{3}} = \left(2^3\right)^{\frac{2}{3}} = 2^2 = 4
\]

So, the equation becomes:

\[
x + 5 = 4
\]

### Step 3: Solve for \(x\).
Now, subtract 5 from both sides:

\[
x = 4 - 5 = -1
\]

Thus, the solution is:

\[
x = -1
\]

### Final Answer:
\[
x = -1
\]

Do you want any more details or clarifications on this solution?

Here are five related questions for further practice:

1. Solve \((x + 3)^{\frac{3}{2}} = 27\).
2. Solve \( (x - 4)^{\frac{2}{3}} = 9 \).
3. What is the value of \( x \) in \((x + 7)^{\frac{3}{2}} = 125\)?
4. Solve \(x^{\frac{5}{2}} = 32\).
5. What is \(x\) if \( (x + 1)^{\frac{2}{5}} = 16\)?

**Tip:** When solving exponents with fractional powers, raising both sides to the reciprocal of the exponent is an effective way to eliminate the power and simplify the equation.

This page provides a detailed step-by-step solution to the equation (x + 5)^(3/2) = 8. Learn how to simplify exponents with fractions and solve algebraic equations efficiently. Suitable for students in Grades 9-10.

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