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The given equation is:

\[
\left( u + 9 \right)^{\frac{4}{5}} = 2
\]

### Step 1: Isolate \( (u + 9) \)
We have:

\[
\left( u + 9 \right)^{\frac{4}{5}} = 2
\]

To eliminate the exponent \(\frac{4}{5}\), we raise both sides of the equation to the reciprocal power, which is \(\frac{5}{4}\):

\[
(u + 9) = 2^{\frac{5}{4}}
\]

### Step 2: Simplify \( 2^{\frac{5}{4}} \)
We can compute \( 2^{\frac{5}{4}} \). First, find \( 2^5 = 32 \), and then take the fourth root:

\[
2^{\frac{5}{4}} = \sqrt[4]{32}
\]

So:

\[
u + 9 = \sqrt[4]{32}
\]

### Step 3: Solve for \( u \)
Now, isolate \( u \) by subtracting 9 from both sides:

\[
u = \sqrt[4]{32} - 9
\]

The fourth root of 32 is approximately \( 2.378 \). So:

\[
u \approx 2.378 - 9
\]

\[
u \approx -6.622
\]

Thus, the solution is approximately:

\[
u \approx -6.622
\]

If you'd like more details or have any questions, feel free to ask!

Here are 5 related questions for further practice:
1. How would you solve \( (x + 3)^{\frac{2}{3}} = 5 \)?
2. What is the reciprocal of a fractional exponent?
3. How do you handle negative exponents in equations?
4. Can you express \( 2^{\frac{3}{2}} \) in its simplest radical form?
5. What is the general solution for \( (a + b)^n = c \)?

**Tip:** When solving fractional exponent equations, remember to always raise both sides to the reciprocal of the fraction to eliminate the exponent!

Solve (u + 9)^(4/5) = 2 where u is a real number. Simplify your answer as much as possible.

Learn how to solve the algebraic equation (u + 9)^(4/5) = 2 using fractional exponents and properties of radicals. This step-by-step guide simplifies the equation and demonstrates how to isolate the variable u. Suitable for high school students in Grades 10-12.

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