Fully simplify the expression below. $$ h^{9} k^{-4} \times 4 h^{6} k^{-5} \div\left(10 h^{2} k^{-3}\right)^{-2} $$ Give your answer as a fraction without any negative indices.

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
h^{9} k^{-4} \times 4 h^{6} k^{-5} \div\left(10 h^{2} k^{-3}\right)^{-2}
$$

**Step-by-Step Simplification:**

1. **Rewrite the Division as Multiplication:**

$$
h^{9} k^{-4} \times 4 h^{6} k^{-5} \times \left(10 h^{2} k^{-3}\right)^{2}
$$

2. **Expand the Exponents:**

$$
\left(10 h^{2} k^{-3}\right)^{2} = 10^{2} h^{4} k^{-6} = 100 h^{4} k^{-6}
$$

3. **Combine All Terms:**

$$
h^{9} k^{-4} \times 4 h^{6} k^{-5} \times 100 h^{4} k^{-6}
$$

4. **Multiply the Coefficients:**

$$
4 \times 100 = 400
$$

5. **Add the Exponents for Like Bases:**

$$
h^{9 + 6 + 4} = h^{19} \
k^{-4 -5 -6} = k^{-15}
$$

6. **Combine Everything:**

$$
400 h^{19} k^{-15}
$$

7. **Express with Positive Exponents:**

$$
\frac{400 h^{19}}{k^{15}}
$$

**Final Simplified Expression:**

$$
\frac{400\, h^{19}}{k^{15}}
$$
$$ \frac{400\, h^{19}}{k^{15}} $$

Fully simplify the expression below.

Give your answer as a fraction without any negative
indices.

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Fully simplify the expression below. Give your answer as a fraction without any negative indices.