Express the following fraction in simplest form, only using positive exponents.

start fraction, minus, 3, left parenthesis, v, cubed, right parenthesis, to the power minus 5 , divided by, 6, s, to the power minus 7 , end fraction

6*s*−7−3(*v*3)−5

To simplify the given expression and express it with only positive exponents, let's start by analyzing the given fraction:

\[
\frac{-3(v^3)^{-5}}{6s^{-7}}
\]

### Step-by-step Simplification:

1. **Simplify the numerator and the denominator:**

   - **Numerator:** \(-3(v^3)^{-5}\)

     Applying the power rule \((a^m)^n = a^{mn}\), we get:

     \[
     (v^3)^{-5} = v^{3 \cdot (-5)} = v^{-15}
     \]

     Therefore, the numerator becomes:

     \[
     -3v^{-15}
     \]

   - **Denominator:** \(6s^{-7}\)

     The denominator remains the same for now.

2. **Rewrite the expression:**

   The fraction now looks like:

   \[
   \frac{-3v^{-15}}{6s^{-7}}
   \]

3. **Combine the coefficients:**

   The coefficients are \(-3\) and \(6\), which simplify as:

   \[
   \frac{-3}{6} = -\frac{1}{2}
   \]

   So the fraction becomes:

   \[
   -\frac{1}{2} \cdot \frac{v^{-15}}{s^{-7}}
   \]

4. **Apply the rule for negative exponents:** 

   Recall that \(a^{-n} = \frac{1}{a^n}\). So:

   - \(v^{-15} = \frac{1}{v^{15}}\)
   - \(s^{-7} = \frac{1}{s^7}\)

   When we divide by \(s^{-7}\), we multiply by \(s^7\). Therefore, we have:

   \[
   -\frac{1}{2} \cdot \frac{s^7}{v^{15}}
   \]

5. **Combine and write in simplest form:**

   This results in:

   \[
   -\frac{s^7}{2v^{15}}
   \]

### Final Answer:

\[
-\frac{s^7}{2v^{15}}
\]

This is the expression in its simplest form, with only positive exponents.

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Would you like more details on any of these steps? Here are some related questions you might be interested in:

1. How do you multiply and divide expressions with exponents?
2. What is the rule for converting negative exponents to positive exponents?
3. How can you simplify complex fractions involving variables and exponents?
4. What are the steps to factor expressions with negative exponents?
5. How do you solve equations that involve negative exponents?

**Tip:** Always remember to convert negative exponents to positive by using their reciprocal, as this is essential for simplifying expressions properly.

Express the following fraction in simplest form, only using positive exponents: start fraction, minus, 3, left parenthesis, v, cubed, right parenthesis, to the power minus 5 , divided by, 6, s, to the power minus 7 , end fraction.

Learn how to simplify the fraction -3(v^3)^-5 / 6s^-7 by applying exponent rules. This detailed solution explains how to handle negative exponents and combine like terms, ideal for students in Grades 9-12.

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