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The expression provided is \((121x^{-8})^{1/2}\).

Let's simplify it step by step:

### Step 1: Apply the exponent rule
When an expression with exponents is raised to a power, we can multiply the exponents. Specifically, we use the rule:
\[
(a^m b^n)^{p} = a^{mp} b^{np}
\]
In this case:
\[
(121x^{-8})^{1/2} = 121^{1/2} \cdot (x^{-8})^{1/2}
\]

### Step 2: Simplify \(121^{1/2}\)
The square root of 121 is 11, so:
\[
121^{1/2} = 11
\]

### Step 3: Simplify \((x^{-8})^{1/2}\)
For the exponent of \(x\), we multiply \(-8\) by \(\frac{1}{2}\), which gives:
\[
x^{-8 \cdot \frac{1}{2}} = x^{-4}
\]

### Step 4: Combine the results
Thus, the expression simplifies to:
\[
11x^{-4}
\]

This can also be written as:
\[
\frac{11}{x^4}
\]

So, the simplified form of \((121x^{-8})^{1/2}\) is \(\frac{11}{x^4}\).

Would you like further details or clarifications on any of the steps?

Here are 5 related questions to deepen your understanding:
1. How would you simplify \((144x^{6})^{1/2}\)?
2. What happens if you have \((a^m b^n)^{p}\) with fractional exponents?
3. Can you rewrite \(x^{-n}\) as a positive exponent?
4. What would \((x^{8} \cdot y^{-4})^{1/2}\) simplify to?
5. How does this simplification change if the exponent were \(\frac{3}{2}\) instead of \(\frac{1}{2}\)?

**Tip:** Always break down complex expressions step by step by applying exponent rules, especially when fractional powers are involved.

Learn how to simplify the expression (121x^{-8})^{1/2} using exponent rules. This step-by-step guide explains the process of simplifying exponents and radicals, focusing on key concepts such as exponent laws and square roots. Suitable for students in Grades 9-11.

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