To simplify the given fraction with positive exponents, let's break it down step by step:

Given expression: $\frac{-3 \left(h^3 t^{-5}\right)^{-3}}{6 h^4}$

Step 1: Simplify the numerator

The numerator is $-3 \left(h^3 t^{-5}\right)^{-3}$.

When we raise a product to a power, we apply the exponent to each factor: $\left(h^3 t^{-5}\right)^{-3} = h^{3(-3)} t^{-5(-3)} = h^{-9} t^{15}$ Now the numerator becomes: $-3 h^{-9} t^{15}$

Step 2: Write the fraction

Now the fraction looks like: $\frac{-3 h^{-9} t^{15}}{6 h^4}$

Step 3: Simplify the terms

• Combine the $h$ terms by subtracting exponents in the fraction: $h^{-9} / h^4 = h^{-9 - 4} = h^{-13}$ Now the fraction is: $\frac{-3 t^{15}}{6 h^{13}}$

Step 4: Simplify the constants

We can simplify the constant term $\frac{-3}{6}$ to $-\frac{1}{2}$, so we have: $-\frac{1}{2} \frac{t^{15}}{h^{13}}$

Final simplified form:

$-\frac{t^{15}}{2 h^{13}}$

This is the simplest form of the given expression, with positive exponents.

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Would you like more details on any of these steps, or do you have another question?

Here are some related questions for further practice:

1. Simplify the expression: $\frac{2x^{-3} y^4}{4x^5 y^{-2}}$.
2. Simplify the expression: $\frac{a^{-2} b^3}{c^4 a^{-1} b^2}$.
3. How would you simplify $\frac{3x^4}{5x^{-3}}$?
4. Express $\frac{m^{-5} n^7}{2m^3 n^{-4}}$ in simplest form.
5. What is the result of simplifying $\frac{4a^3 b^{-2}}{2a^{-1} b^5}$?

Tip: Always remember that when dividing powers with the same base, you subtract the exponents.