Simplify p(p^-7q^3)^-2q^-3

$p {(p^{- 7} q^{3})}^{- 2} q^{- 3}$

Step 1

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

$p {(\frac{1}{p^{7}} q^{3})}^{- 2} q^{- 3}$

Step 2

Combine $\frac{1}{p^{7}}$ and $q^{3}$ .

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

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 4

Apply the product rule to $\frac{p^{7}}{q^{3}}$ .

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

Step 5

Multiply the exponents in ${(p^{7})}^{2}$ .

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

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

Multiply $7$ by $2$ .

$p \frac{p^{14}}{{(q^{3})}^{2}} q^{- 3}$

Step 6

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

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

$p \frac{p^{14}}{q^{3 \cdot 2}} q^{- 3}$

Multiply $3$ by $2$ .

$p \frac{p^{14}}{q^{6}} q^{- 3}$

Step 7

Multiply $p \frac{p^{14}}{q^{6}}$ .

Combine $p$ and $\frac{p^{14}}{q^{6}}$ .

$\frac{p \cdot p^{14}}{q^{6}} q^{- 3}$

Multiply $p$ by $p^{14}$ by adding the exponents.

Multiply $p$ by $p^{14}$ .

Raise $p$ to the power of $1$ .

$\frac{p^{1} p^{14}}{q^{6}} q^{- 3}$

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

$\frac{p^{1 + 14}}{q^{6}} q^{- 3}$

Add $1$ and $14$ .

$\frac{p^{15}}{q^{6}} q^{- 3}$

Step 8

Cancel the common factor of $q^{- 3}$ .

Factor $q^{- 3}$ out of $q^{6}$ .

$\frac{p^{15}}{q^{- 3} q^{9}} q^{- 3}$

Cancel the common factor.

$\frac{p^{15}}{q^{- 3} q^{9}} q^{- 3}$

Rewrite the expression.

$\frac{p^{15}}{q^{9}}$