how to solve ($$ \placeholder{} $$8x-2)$$ ^{\frac32}\text{ }=\text{ }27 $$

To solve the equation $ (8x - 2)^{\frac{3}{2}} = 27 $, follow these steps:

### Step 1: Isolate the power expression
The equation is already in a form where the term $ (8x - 2) $ is raised to the power $ \frac{3}{2} $, so start by eliminating the exponent.

To do this, raise both sides of the equation to the power $ \frac{2}{3} $ to cancel out the $ \frac{3}{2} $ exponent:

\[
\left( (8x - 2)^{\frac{3}{2}} \right)^{\frac{2}{3}} = 27^{\frac{2}{3}}
\]

Since $ \left( (8x - 2)^{\frac{3}{2}} \right)^{\frac{2}{3}} = 8x - 2 $, you get:

\[
8x - 2 = 27^{\frac{2}{3}}
\]

### Step 2: Simplify $ 27^{\frac{2}{3}} $
Now simplify the right-hand side:

\[
27^{\frac{2}{3}} = \left( 3^3 \right)^{\frac{2}{3}} = 3^2 = 9
\]

This gives:

\[
8x - 2 = 9
\]

### Step 3: Solve for $ x $
Now solve the linear equation for $ x $:

\[
8x = 9 + 2
\]

\[
8x = 11
\]

\[
x = \frac{11}{8}
\]

Thus, the solution is:

\[
x = \frac{11}{8}
\]

### Final Answer:
\[
x = \frac{11}{8}
\]

---

Would you like more details or have any questions about this solution?

Here are 5 related questions you might find interesting:
1. How do you solve $ (5x + 1)^{\frac{2}{3}} = 16 $?
2. What are the properties of rational exponents?
3. How can you solve equations with higher powers, such as $ (3x - 4)^4 = 81 $?
4. How do you solve logarithmic equations involving exponents?
5. Can you explain the meaning of fractional exponents in more detail?

**Tip**: When dealing with fractional exponents, remember that the numerator indicates the power, and the denominator indicates the root.

Learn how to solve the equation (8x - 2)^{3/2} = 27 step-by-step. This solution covers the properties of rational exponents and how to isolate the variable. Suitable for students in Grades 9-11.

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