how to step by step simplify (9x with an exponent of 2 y with an exponent of 8) and all of that with an exponent of 3/2.

To simplify \((9x^2 y^8)^{\frac{3}{2}}\), follow these steps:

### Step 1: Apply the power rule \((a^m)^n = a^{mn}\)
When raising a product to an exponent, we apply the exponent to each factor inside the parentheses.

\[
(9x^2 y^8)^{\frac{3}{2}} = 9^{\frac{3}{2}} \cdot (x^2)^{\frac{3}{2}} \cdot (y^8)^{\frac{3}{2}}
\]

### Step 2: Simplify each term individually

#### 1. Simplify \(9^{\frac{3}{2}}\)
The exponent \(\frac{3}{2}\) can be broken down as:
\[
9^{\frac{3}{2}} = \left(9^{\frac{1}{2}}\right)^3 = (3)^3 = 27
\]

#### 2. Simplify \((x^2)^{\frac{3}{2}}\)
Using the rule \((a^m)^n = a^{m \cdot n}\):
\[
(x^2)^{\frac{3}{2}} = x^{2 \cdot \frac{3}{2}} = x^3
\]

#### 3. Simplify \((y^8)^{\frac{3}{2}}\)
Similarly, apply the power rule:
\[
(y^8)^{\frac{3}{2}} = y^{8 \cdot \frac{3}{2}} = y^{12}
\]

### Step 3: Multiply all the simplified terms
Now, combine all the simplified components:
\[
27 \cdot x^3 \cdot y^{12}
\]

Thus, the simplified expression is:

\[
27x^3y^{12}
\]

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Would you like further details on any of the steps? Here are 5 relative questions you might explore:

1. How do you simplify more complex exponents, such as fractional exponents of negative numbers?
2. What are the rules for raising a product or quotient to a power?
3. How can you apply similar methods to expressions with roots, like \((\sqrt{x^4})^{\frac{3}{2}}\)?
4. What is the connection between fractional exponents and radicals?
5. How can you simplify more complex algebraic expressions involving multiple variables?

**Tip**: Always break down the exponents step by step, especially when dealing with fractional powers, to avoid errors.

Follow a detailed step-by-step process to simplify (9x^2 y^8)^{3/2} using key exponentiation rules. This problem explains the power rule for algebraic expressions, fractional exponents, and simplifying powers of variables. Perfect for high school students in Grades 9-12.

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