Simplify ((4p^-4q)^-2)/(10pq^-3)

$\frac{{(4 p^{- 4} q)}^{- 2}}{10 p q^{- 3}}$

Step 1

Simplify the expression.

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

$\frac{1}{10 p q^{- 3} {(4 p^{- 4} q)}^{2}}$

Move $q^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{q^{3}}{10 p {(4 p^{- 4} q)}^{2}}$

Step 2

Simplify the denominator.

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

$\frac{q^{3}}{10 p {(4 p^{- 4})}^{2} q^{2}}$

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

$\frac{q^{3}}{10 p {(4 \frac{1}{p^{4}})}^{2} q^{2}}$

Combine $4$ and $\frac{1}{p^{4}}$ .

$\frac{q^{3}}{10 p {(\frac{4}{p^{4}})}^{2} q^{2}}$

Apply the product rule to $\frac{4}{p^{4}}$ .

$\frac{q^{3}}{10 p \frac{4^{2}}{{(p^{4})}^{2}} q^{2}}$

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

$\frac{q^{3}}{10 p \frac{16}{{(p^{4})}^{2}} q^{2}}$

Multiply the exponents in ${(p^{4})}^{2}$ .

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

$\frac{q^{3}}{10 p \frac{16}{p^{4 \cdot 2}} q^{2}}$

Multiply $4$ by $2$ .

$\frac{q^{3}}{10 p \frac{16}{p^{8}} q^{2}}$

Combine exponents.

Combine $10$ and $\frac{16}{p^{8}}$ .

$\frac{q^{3}}{p \frac{10 \cdot 16}{p^{8}} q^{2}}$

Multiply $10$ by $16$ .

$\frac{q^{3}}{p \frac{160}{p^{8}} q^{2}}$

Combine $p$ and $\frac{160}{p^{8}}$ .

$\frac{q^{3}}{\frac{p \cdot 160}{p^{8}} q^{2}}$

Combine $\frac{p \cdot 160}{p^{8}}$ and $q^{2}$ .

$\frac{q^{3}}{\frac{p \cdot 160 q^{2}}{p^{8}}}$

Reduce the expression $\frac{p \cdot 160 q^{2}}{p^{8}}$ by cancelling the common factors.

Factor $p$ out of $p \cdot 160 q^{2}$ .

$\frac{q^{3}}{\frac{p (160 q^{2})}{p^{8}}}$

Factor $p$ out of $p^{8}$ .

$\frac{q^{3}}{\frac{p (160 q^{2})}{p \cdot p^{7}}}$

Cancel the common factor.

$\frac{q^{3}}{\frac{p (160 q^{2})}{p \cdot p^{7}}}$

Rewrite the expression.

$\frac{q^{3}}{\frac{160 q^{2}}{p^{7}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$q^{3} \frac{p^{7}}{160 q^{2}}$

Step 4

Cancel the common factor of $q^{2}$ .

Factor $q^{2}$ out of $q^{3}$ .

$q^{2} q \frac{p^{7}}{160 q^{2}}$

Factor $q^{2}$ out of $160 q^{2}$ .

$q^{2} q \frac{p^{7}}{q^{2} \cdot 160}$

Cancel the common factor.

$q^{2} q \frac{p^{7}}{q^{2} \cdot 160}$

Rewrite the expression.

$q \frac{p^{7}}{160}$

Step 5

Combine $q$ and $\frac{p^{7}}{160}$ .

$\frac{q p^{7}}{160}$

Simplify ((4p^-4q)^-2)/(10pq^-3)

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