Simplify ((4qh^-3)^-2)/(12q^8h^8)

$\frac{{(4 q h^{- 3})}^{- 2}}{12 q^{8} h^{8}}$

Step 1

Move ${(4 q h^{- 3})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{12 q^{8} h^{8} {(4 q h^{- 3})}^{2}}$

Step 2

Simplify the denominator.

Apply the product rule to $4 q h^{- 3}$ .

$\frac{1}{12 q^{8} h^{8} {(4 q)}^{2} {(h^{- 3})}^{2}}$

Apply the product rule to $4 q$ .

$\frac{1}{12 q^{8} h^{8} (4^{2} q^{2}) {(h^{- 3})}^{2}}$

Raise $4$ to the power of $2$ .

$\frac{1}{12 q^{8} h^{8} (16 q^{2}) {(h^{- 3})}^{2}}$

Multiply the exponents in ${(h^{- 3})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{12 q^{8} h^{8} (16 q^{2}) h^{- 3 \cdot 2}}$

Multiply $- 3$ by $2$ .

$\frac{1}{12 q^{8} h^{8} (16 q^{2}) h^{- 6}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{12 q^{8} h^{8} \cdot 16 q^{2} \frac{1}{h^{6}}}$

Combine exponents.

Multiply $16$ by $12$ .

$\frac{1}{192 q^{8} h^{8} q^{2} \frac{1}{h^{6}}}$

Multiply $q^{8}$ by $q^{2}$ by adding the exponents.

Move $q^{2}$ .

$\frac{1}{192 (q^{2} q^{8}) h^{8} \frac{1}{h^{6}}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{192 q^{2 + 8} h^{8} \frac{1}{h^{6}}}$

Add $2$ and $8$ .

$\frac{1}{192 q^{10} h^{8} \frac{1}{h^{6}}}$

Combine $192$ and $\frac{1}{h^{6}}$ .

$\frac{1}{q^{10} h^{8} \frac{192}{h^{6}}}$

Combine $q^{10}$ and $\frac{192}{h^{6}}$ .

$\frac{1}{h^{8} \frac{q^{10} \cdot 192}{h^{6}}}$

Combine $h^{8}$ and $\frac{q^{10} \cdot 192}{h^{6}}$ .

$\frac{1}{\frac{h^{8} (q^{10} \cdot 192)}{h^{6}}}$

Remove unnecessary parentheses.

$\frac{1}{\frac{h^{8} q^{10} \cdot 192}{h^{6}}}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{h^{8} q^{10} \cdot 192}{h^{6}}$ by cancelling the common factors.

Factor $h^{6}$ out of $h^{8} q^{10} \cdot 192$ .

$\frac{1}{\frac{h^{6} (h^{2} q^{10} \cdot 192)}{h^{6}}}$

Multiply by $1$ .

$\frac{1}{\frac{h^{6} (h^{2} q^{10} \cdot 192)}{h^{6} \cdot 1}}$

Cancel the common factor.

$\frac{1}{\frac{h^{6} (h^{2} q^{10} \cdot 192)}{h^{6} \cdot 1}}$

Rewrite the expression.

$\frac{1}{\frac{h^{2} q^{10} \cdot 192}{1}}$

Divide $h^{2} q^{10} \cdot 192$ by $1$ .

$\frac{1}{h^{2} q^{10} \cdot 192}$

Step 3

Move $192$ to the left of $h^{2} q^{10}$ .

$\frac{1}{192 h^{2} q^{10}}$

Simplify ((4qh^-3)^-2)/(12q^8h^8)

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