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Simplify the expression. Express the answer using positive exponent.

The expression provided is:

\[
\left(\frac{6r^7 s^v}{7r^1 s^5}\right)^2
\]

### Step 1: Simplify the expression inside the parentheses.
1. **Simplify the coefficients:**
   - The coefficient \( \frac{6}{7} \) remains as it is because there are no common factors between 6 and 7.

2. **Simplify the exponents for \( r \):**
   - \( r^7 \) divided by \( r^1 \) simplifies to \( r^{7-1} = r^6 \).

3. **Simplify the exponents for \( s \):**
   - \( s^v \) divided by \( s^5 \) simplifies to \( s^{v-5} \).

So the simplified expression inside the parentheses is:

\[
\frac{6r^6 s^{v-5}}{7}
\]

### Step 2: Raise the simplified expression to the power of 2.
Apply the power of 2 to each term in the expression:

\[
\left(\frac{6r^6 s^{v-5}}{7}\right)^2 = \frac{6^2 \cdot r^{6 \times 2} \cdot s^{(v-5) \times 2}}{7^2}
\]

This simplifies to:

\[
\frac{36r^{12} s^{2(v-5)}}{49}
\]

### Step 3: Express the answer using positive exponents.
Since all exponents are already positive, the final simplified expression is:

\[
\frac{36r^{12} s^{2(v-5)}}{49}
\]

### Summary:
The simplified expression is \( \frac{36r^{12} s^{2(v-5)}}{49} \).

Would you like any further details or have any other questions?

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### Related Questions:
1. How does the exponent outside the parentheses affect each term inside the expression?
2. What happens if \( v = 5 \) in the given expression?
3. Can you explain why the coefficient 6 was squared, but the variables were raised to the power of 2?
4. How would the expression change if one of the exponents inside the fraction were negative?
5. What other rules of exponents are important to remember when simplifying complex fractions?

### Tip:
When working with fractions involving exponents, always simplify the expression inside the parentheses first before applying the external exponent. This will help avoid errors in calculations.

Learn how to simplify the exponential expression \( \left(\frac{6r^7 s^v}{7r^1 s^5}\right)^2 \) using positive exponents. Step-by-step solution provided, covering exponents and fraction simplification. Suitable for Grades 7-9 students.

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