Simplify (a^3b^5c^-4)^2

${(a^{3} b^{5} c^{- 4})}^{2}$

Step 1

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

${(a^{3} b^{5} \frac{1}{c^{4}})}^{2}$

Step 2

Multiply $a^{3} b^{5} \frac{1}{c^{4}}$ .

Combine $a^{3}$ and $\frac{1}{c^{4}}$ .

${(b^{5} \frac{a^{3}}{c^{4}})}^{2}$

Combine $b^{5}$ and $\frac{a^{3}}{c^{4}}$ .

${(\frac{b^{5} a^{3}}{c^{4}})}^{2}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{b^{5} a^{3}}{c^{4}}$ .

$\frac{{(b^{5} a^{3})}^{2}}{{(c^{4})}^{2}}$

Apply the product rule to $b^{5} a^{3}$ .

$\frac{{(b^{5})}^{2} {(a^{3})}^{2}}{{(c^{4})}^{2}}$

Step 4

Simplify the numerator.

Multiply the exponents in ${(b^{5})}^{2}$ .

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

$\frac{b^{5 \cdot 2} {(a^{3})}^{2}}{{(c^{4})}^{2}}$

Multiply $5$ by $2$ .

$\frac{b^{10} {(a^{3})}^{2}}{{(c^{4})}^{2}}$

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

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

$\frac{b^{10} a^{3 \cdot 2}}{{(c^{4})}^{2}}$

Multiply $3$ by $2$ .

$\frac{b^{10} a^{6}}{{(c^{4})}^{2}}$

Step 5

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

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

$\frac{b^{10} a^{6}}{c^{4 \cdot 2}}$

Multiply $4$ by $2$ .

$\frac{b^{10} a^{6}}{c^{8}}$