To solve the equation \((x + 5)^{\frac{2}{3}} = 4\), follow these steps:

### Step 1: Isolate the expression inside the exponent.
The equation is already isolated in the form \((x + 5)^{\frac{2}{3}} = 4\).

### Step 2: Eliminate the fractional exponent.
To eliminate the exponent \(\frac{2}{3}\), raise both sides of the equation to the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This step undoes the exponent on the left-hand side:

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

On the left-hand side, the exponents cancel out, leaving:

\[
x + 5 = 4^{\frac{3}{2}}
\]

### Step 3: Simplify \(4^{\frac{3}{2}}\).
First, break down \(4^{\frac{3}{2}}\) as follows:

\[
4^{\frac{3}{2}} = \left(4^{\frac{1}{2}}\right)^3 = (2)^3 = 8
\]

Thus, the equation becomes:

\[
x + 5 = 8
\]

### Step 4: Solve for \(x\).
Now, subtract 5 from both sides:

\[
x = 8 - 5
\]
\[
x = 3
\]

### Final Answer:
\[
x = 3
\]

Let me know if you'd like more details, or have any questions!

Here are 5 related questions you might find useful:
1. What is the general method for solving equations with fractional exponents?
2. How do you solve radical equations?
3. What happens if the exponent is negative in a similar problem?
4. How do you handle an equation like \((x + 7)^{\frac{1}{4}} = 3\)?
5. Can fractional exponents be expressed as radicals?

**Tip:** Fractional exponents can always be rewritten as roots or powers, making it easier to manipulate and solve equations involving them.

Learn how to solve the equation (x+5)^{2/3} = 4 using exponentiation techniques. This step-by-step solution breaks down handling fractional exponents, simplifying the powers, and solving for x. Ideal for students in Grades 9-11.

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