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Fully simplify the expression below.

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

Give your answer as a fraction without any negative indices.


Sagot :

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]