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:

$x = \frac{11}{8}$

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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.