Simplify (-5(m^-4n^-5)^-3)/(7(p^-6q^8)^-4)

$\frac{- 5 {(m^{- 4} n^{- 5})}^{- 3}}{7 {(p^{- 6} q^{8})}^{- 4}}$

Step 1

Simplify.

Move ${(m^{- 4} n^{- 5})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 5}{7 {(p^{- 6} q^{8})}^{- 4} {(m^{- 4} n^{- 5})}^{3}}$

Move ${(p^{- 6} q^{8})}^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{- 5 {(p^{- 6} q^{8})}^{4}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Step 2

Simplify the numerator.

Apply the product rule to $p^{- 6} q^{8}$ .

$\frac{- 5 {(p^{- 6})}^{4} {(q^{8})}^{4}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Multiply the exponents in ${(p^{- 6})}^{4}$ .

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

$\frac{- 5 p^{- 6 \cdot 4} {(q^{8})}^{4}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Multiply $- 6$ by $4$ .

$\frac{- 5 p^{- 24} {(q^{8})}^{4}}{7 {(m^{- 4} n^{- 5})}^{3}}$

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

$\frac{- 5 \frac{1}{p^{24}} {(q^{8})}^{4}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Multiply the exponents in ${(q^{8})}^{4}$ .

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

$\frac{- 5 \frac{1}{p^{24}} q^{8 \cdot 4}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Multiply $8$ by $4$ .

$\frac{- 5 \frac{1}{p^{24}} q^{32}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Combine exponents.

Combine $- 5$ and $\frac{1}{p^{24}}$ .

$\frac{\frac{- 5}{p^{24}} q^{32}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Combine $\frac{- 5}{p^{24}}$ and $q^{32}$ .

$\frac{\frac{- 5 q^{32}}{p^{24}}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Move the negative in front of the fraction.

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 {(m^{- 4} n^{- 5})}^{3}}$

Step 3

Simplify the denominator.

Apply the product rule to $m^{- 4} n^{- 5}$ .

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 {(m^{- 4})}^{3} {(n^{- 5})}^{3}}$

Multiply the exponents in ${(m^{- 4})}^{3}$ .

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

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 m^{- 4 \cdot 3} {(n^{- 5})}^{3}}$

Multiply $- 4$ by $3$ .

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 m^{- 12} {(n^{- 5})}^{3}}$

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

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 \frac{1}{m^{12}} {(n^{- 5})}^{3}}$

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

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

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 \frac{1}{m^{12}} n^{- 5 \cdot 3}}$

Multiply $- 5$ by $3$ .

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 \frac{1}{m^{12}} n^{- 15}}$

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

$\frac{- \frac{5 q^{32}}{p^{24}}}{7 \frac{1}{m^{12}} \frac{1}{n^{15}}}$

Combine exponents.

Combine $7$ and $\frac{1}{m^{12}}$ .

$\frac{- \frac{5 q^{32}}{p^{24}}}{\frac{7}{m^{12}} \cdot \frac{1}{n^{15}}}$

Multiply $\frac{7}{m^{12}}$ by $\frac{1}{n^{15}}$ .

$\frac{- \frac{5 q^{32}}{p^{24}}}{\frac{7}{m^{12} n^{15}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{5 q^{32}}{p^{24}} \cdot \frac{m^{12} n^{15}}{7}$

Step 5

Multiply $\frac{m^{12} n^{15}}{7}$ by $\frac{5 q^{32}}{p^{24}}$ .

$- \frac{m^{12} n^{15} (5 q^{32})}{7 p^{24}}$

Step 6

Move $5$ to the left of $m^{12} n^{15}$ .

$- \frac{5 m^{12} n^{15} q^{32}}{7 p^{24}}$

Simplify (-5(m^-4n^-5)^-3)/(7(p^-6q^8)^-4)

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