The expression shown in the image is:

$\frac{-3v^{-8}}{-2(v^{-2})^3}$

We are tasked with simplifying this expression using only positive exponents. Let's work through the simplification step by step.

Step 1: Simplify the denominator

The denominator has $(v^{-2})^3$. Apply the exponent rule $(a^m)^n = a^{m \cdot n}$ to get: $(v^{-2})^3 = v^{-6}$

Thus, the denominator becomes: $-2v^{-6}$

Step 2: Rewrite the expression

Now the expression looks like: $\frac{-3v^{-8}}{-2v^{-6}}$

Step 3: Simplify the fraction

The negative signs cancel out, so we now have: $\frac{3v^{-8}}{2v^{-6}}$

Next, use the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ for the $v$ terms: $\frac{v^{-8}}{v^{-6}} = v^{-8 - (-6)} = v^{-2}$

So the expression simplifies to: $\frac{3}{2}v^{-2}$

Step 4: Express with positive exponents

To express $v^{-2}$ with a positive exponent, rewrite it as: $\frac{3}{2} \cdot \frac{1}{v^2} = \frac{3}{2v^2}$

Final Answer:

The simplified form of the given expression is: $\frac{3}{2v^2}$

Would you like a further explanation or have any other questions? Here are some related questions to consider:

1. What are the rules for handling negative exponents?
2. How do you apply the power of a power rule in exponentiation?
3. What happens when dividing terms with the same base but different exponents?
4. How do you simplify expressions with multiple variables and exponents?
5. What are some other methods to simplify fractions involving exponents?

Tip: When simplifying exponents, always remember to combine powers carefully and rewrite negative exponents as positive using reciprocals.