Simplify (1/2a^3b^4c^5)^3

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

Step 1

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

Combine $a^{3}$ and $\frac{1}{2}$ .

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

Combine $b^{4}$ and $\frac{a^{3}}{2}$ .

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

Combine $\frac{b^{4} a^{3}}{2}$ and $c^{5}$ .

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

Step 2

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

Apply the product rule to $\frac{b^{4} a^{3} c^{5}}{2}$ .

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

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

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

Apply the product rule to $b^{4} a^{3}$ .

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

Step 3

Simplify the numerator.

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

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

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

Multiply $4$ by $3$ .

$\frac{b^{12} {(a^{3})}^{3} {(c^{5})}^{3}}{2^{3}}$

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

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

$\frac{b^{12} a^{3 \cdot 3} {(c^{5})}^{3}}{2^{3}}$

Multiply $3$ by $3$ .

$\frac{b^{12} a^{9} {(c^{5})}^{3}}{2^{3}}$

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

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

$\frac{b^{12} a^{9} c^{5 \cdot 3}}{2^{3}}$

Multiply $5$ by $3$ .

$\frac{b^{12} a^{9} c^{15}}{2^{3}}$

Step 4

Raise $2$ to the power of $3$ .

$\frac{b^{12} a^{9} c^{15}}{8}$