Certainly! Let's simplify the given expression step by step.
Given expression:
[tex]\[
h^9 k^{-4} \times 4 h^6 k^{-5} \div \left(10 h^2 k^{-3}\right)^{-2}
\][/tex]
### Step 1: Simplify the denominator [tex]\((10 h^2 k^{-3})^{-2}\)[/tex]
When we apply the exponent [tex]\(-2\)[/tex] to the entire term inside the parentheses, we need to distribute [tex]\(-2\)[/tex] to each factor inside.
[tex]\[
(10 h^2 k^{-3})^{-2} = 10^{-2} h^{2 \cdot -2} k^{-3 \cdot -2}
\][/tex]
This simplifies to:
[tex]\[
10^{-2} h^{-4} k^6
\][/tex]
### Step 2: Rewrite the original expression using this simplified denominator
The original expression now looks like:
[tex]\[
h^9 k^{-4} \times 4 h^6 k^{-5} \times 10^{-2} h^{-4} k^6
\][/tex]
### Step 3: Combine the like terms
We need to combine the coefficients and the exponents of like bases (i.e., [tex]\(h\)[/tex] terms and [tex]\(k\)[/tex] terms).
### Coefficients:
The coefficients are [tex]\(1\)[/tex] (from [tex]\(h^9 k^{-4}\)[/tex]), [tex]\(4\)[/tex] (from [tex]\(4 h^6 k^{-5}\)[/tex]), and [tex]\(10^{-2}\)[/tex]:
[tex]\[
1 \times 4 \times 10^{-2} = 4 \times \frac{1}{100} = \frac{4}{100} = \frac{1}{25}
\][/tex]
### Exponents of [tex]\(h\)[/tex]:
Combine the exponents of [tex]\(h\)[/tex]:
[tex]\[
h^{9 + 6 - 4} = h^{11}
\][/tex]
### Exponents of [tex]\(k\)[/tex]:
Combine the exponents of [tex]\(k\)[/tex]:
[tex]\[
k^{-4 - 5 + 6} = k^{-3}
\][/tex]
### Step 4: Construct the final simplified expression
Putting it all together, we have:
[tex]\[
\frac{1}{25} \times h^{11} \times k^{-3} = \frac{h^{11}}{25 k^3}
\][/tex]
Thus, the fully simplified expression is:
[tex]\[
\boxed{\frac{h^{11}}{25 k^3}}
\][/tex]
