https://file.aiquickdraw.com/mathbot/file/1729626290567djcvgl1u.jpg

The given mathematical expression is:

\[
\sqrt{\frac{1}{x^8}}
\]

### Simplifying the expression:

We can rewrite the square root as an exponent:

\[
\sqrt{\frac{1}{x^8}} = \left(\frac{1}{x^8}\right)^{\frac{1}{2}}
\]

Now apply the exponent to both the numerator and denominator:

\[
\left(\frac{1}{x^8}\right)^{\frac{1}{2}} = \frac{1^{\frac{1}{2}}}{x^{8 \times \frac{1}{2}}} = \frac{1}{x^4}
\]

Thus, the simplified expression is:

\[
\frac{1}{x^4}
\]

Would you like further details or have any questions?

Here are some related questions to extend your understanding:

1. How would you simplify \(\sqrt{\frac{1}{x^{12}}}\)?
2. What happens if you take the cube root instead of the square root in the original expression?
3. How would you simplify \(\frac{1}{\sqrt{x^8}}\)?
4. How does the exponent rule \((a^m)^n = a^{mn}\) apply in this case?
5. What if \(x\) were negative? How would that affect the simplification?

**Tip:** When simplifying expressions involving roots and exponents, always consider rewriting roots as fractional exponents for easier manipulation!

Learn how to simplify the square root of 1 over x^8 using exponent rules. This problem focuses on algebraic concepts, radicals, and exponents, ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation