To express \(\frac{1}{7}x^{-\frac{3}{5}}\) using a square root, you need to handle the fractional exponent properly. 

Let's rewrite the expression step-by-step:

1. Start with the original expression:
   \[
   \frac{1}{7}x^{-\frac{3}{5}}
   \]

2. Recall that a negative exponent indicates a reciprocal:
   \[
   x^{-\frac{3}{5}} = \frac{1}{x^{\frac{3}{5}}}
   \]

3. Substitute this into the original expression:
   \[
   \frac{1}{7} \cdot \frac{1}{x^{\frac{3}{5}}} = \frac{1}{7x^{\frac{3}{5}}}
   \]

4. To express \(x^{\frac{3}{5}}\) using a square root, remember that \(x^{\frac{3}{5}} = \left(x^{\frac{1}{5}}\right)^3\). Additionally, \(x^{\frac{1}{5}}\) can be written as the fifth root of \(x\):
   \[
   x^{\frac{3}{5}} = \left( \sqrt[5]{x} \right)^3
   \]

5. Substitute this back into the simplified fraction:
   \[
   \frac{1}{7x^{\frac{3}{5}}} = \frac{1}{7 \left( \sqrt[5]{x} \right)^3}
   \]

6. If you want to incorporate the square root symbol directly into the expression, note that it applies to the fifth root, and you would write it as:
   \[
   \frac{1}{7 \left( \sqrt[5]{x} \right)^3}
   \]

So, the final expression using the square root sign is:
\[
\frac{1}{7 \left( \sqrt[5]{x} \right)^3}
\]

Would you like more details on any of the steps? Do you have any questions?

Here are some related questions to expand on this topic:

1. How do you express other fractional exponents using roots?
2. What is the difference between square roots and other types of roots?
3. How do you simplify expressions with roots and fractional exponents?
4. Can you provide examples of simplifying expressions with different fractional exponents?
5. How do you multiply and divide expressions with roots?
6. What is the relationship between exponents and logarithms?
7. How would you simplify an expression like \(\left( \sqrt[5]{x} \right)^3\) without using exponents?
8. How do you solve equations involving fractional exponents and roots?

**Tip:** When converting between exponents and roots, remember that \(x^{1/n}\) is the \(n\)th root of \(x\), and \(\left(x^{a/b}\right)\) can be expressed as \((\sqrt[b]{x})^a\).

Learn how to express the mathematical expression 1/7x^-3/5 using a square root sign. Follow a detailed step-by-step guide including converting fractional exponents to roots and simplifying expressions. Suitable for high school students and those studying advanced algebra.

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