Simplify (12x^-2)/((3p^-1x^-5)^-3)

$\frac{12 x^{- 2}}{{(3 p^{- 1} x^{- 5})}^{- 3}}$

Simplify.

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12}{{(3 p^{- 1} x^{- 5})}^{- 3} x^{2}}$

Move ${(3 p^{- 1} x^{- 5})}^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{12 {(3 p^{- 1} x^{- 5})}^{3}}{x^{2}}$

Simplify the numerator.

Apply the product rule to $3 p^{- 1} x^{- 5}$ .

$\frac{12 {(3 p^{- 1})}^{3} {(x^{- 5})}^{3}}{x^{2}}$

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

$\frac{12 {(3 \frac{1}{p})}^{3} {(x^{- 5})}^{3}}{x^{2}}$

Combine $3$ and $\frac{1}{p}$ .

$\frac{12 {(\frac{3}{p})}^{3} {(x^{- 5})}^{3}}{x^{2}}$

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

$\frac{12 \frac{3^{3}}{p^{3}} {(x^{- 5})}^{3}}{x^{2}}$

Raise $3$ to the power of $3$ .

$\frac{12 \frac{27}{p^{3}} {(x^{- 5})}^{3}}{x^{2}}$

Multiply the exponents in ${(x^{- 5})}^{3}$ .

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

$\frac{12 \frac{27}{p^{3}} x^{- 5 \cdot 3}}{x^{2}}$

Multiply $- 5$ by $3$ .

$\frac{12 \frac{27}{p^{3}} x^{- 15}}{x^{2}}$

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

$\frac{12 \frac{27}{p^{3}} \frac{1}{x^{15}}}{x^{2}}$

Combine exponents.

Combine $12$ and $\frac{27}{p^{3}}$ .

$\frac{\frac{12 \cdot 27}{p^{3}} \cdot \frac{1}{x^{15}}}{x^{2}}$

Multiply $12$ by $27$ .

$\frac{\frac{324}{p^{3}} \cdot \frac{1}{x^{15}}}{x^{2}}$

Multiply $\frac{324}{p^{3}}$ by $\frac{1}{x^{15}}$ .

$\frac{\frac{324}{p^{3} x^{15}}}{x^{2}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{324}{p^{3} x^{15}} \cdot \frac{1}{x^{2}}$

Combine.

$\frac{324 \cdot 1}{p^{3} x^{15} x^{2}}$

Multiply $x^{15}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$\frac{324 \cdot 1}{p^{3} (x^{2} x^{15})}$

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

$\frac{324 \cdot 1}{p^{3} x^{2 + 15}}$

Add $2$ and $15$ .

$\frac{324 \cdot 1}{p^{3} x^{17}}$

Multiply $324$ by $1$ .

$\frac{324}{p^{3} x^{17}}$