Simplify ((2a^3b^2c)^-3)/((3a^2b^3c)^2)

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

Move ${(2 a^{3} b^{2} c)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Simplify the denominator.

Apply the product rule to $3 a^{2} b^{3} c$ .

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

Apply the product rule to $2 a^{3} b^{2} c$ .

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

Multiply $c^{2}$ by $c^{3}$ by adding the exponents.

Move $c^{3}$ .

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

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

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

Add $3$ and $2$ .

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

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

Apply the product rule to $3 a^{2} b^{3}$ .

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

Apply the product rule to $3 a^{2}$ .

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

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

$\frac{1}{9 {(a^{2})}^{2} {(b^{3})}^{2} c^{5} {(2 a^{3} b^{2})}^{3}}$

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

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

$\frac{1}{9 a^{2 \cdot 2} {(b^{3})}^{2} c^{5} {(2 a^{3} b^{2})}^{3}}$

Multiply $2$ by $2$ .

$\frac{1}{9 a^{4} {(b^{3})}^{2} c^{5} {(2 a^{3} b^{2})}^{3}}$

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

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

$\frac{1}{9 a^{4} b^{3 \cdot 2} c^{5} {(2 a^{3} b^{2})}^{3}}$

Multiply $3$ by $2$ .

$\frac{1}{9 a^{4} b^{6} c^{5} {(2 a^{3} b^{2})}^{3}}$

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

Apply the product rule to $2 a^{3} b^{2}$ .

$\frac{1}{9 a^{4} b^{6} c^{5} ({(2 a^{3})}^{3} {(b^{2})}^{3})}$

Apply the product rule to $2 a^{3}$ .

$\frac{1}{9 a^{4} b^{6} c^{5} (2^{3} {(a^{3})}^{3} {(b^{2})}^{3})}$

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

$\frac{1}{9 a^{4} b^{6} c^{5} (8 {(a^{3})}^{3} {(b^{2})}^{3})}$

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

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

$\frac{1}{9 a^{4} b^{6} c^{5} (8 a^{3 \cdot 3} {(b^{2})}^{3})}$

Multiply $3$ by $3$ .

$\frac{1}{9 a^{4} b^{6} c^{5} (8 a^{9} {(b^{2})}^{3})}$

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

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

$\frac{1}{9 a^{4} b^{6} c^{5} (8 a^{9} b^{2 \cdot 3})}$

Multiply $2$ by $3$ .

$\frac{1}{9 a^{4} b^{6} c^{5} (8 a^{9} b^{6})}$

$\frac{1}{9 a^{4} b^{6} c^{5} \cdot 8 a^{9} b^{6}}$

Combine exponents.

Multiply $8$ by $9$ .

$\frac{1}{72 a^{4} b^{6} c^{5} a^{9} b^{6}}$

Multiply $a^{4}$ by $a^{9}$ by adding the exponents.

Move $a^{9}$ .

$\frac{1}{72 (a^{9} a^{4}) b^{6} c^{5} b^{6}}$

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

$\frac{1}{72 a^{9 + 4} b^{6} c^{5} b^{6}}$

Add $9$ and $4$ .

$\frac{1}{72 a^{13} b^{6} c^{5} b^{6}}$

Multiply $b^{6}$ by $b^{6}$ by adding the exponents.

Move $b^{6}$ .

$\frac{1}{72 a^{13} (b^{6} b^{6}) c^{5}}$

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

$\frac{1}{72 a^{13} b^{6 + 6} c^{5}}$

Add $6$ and $6$ .

$\frac{1}{72 a^{13} b^{12} c^{5}}$

Simplify ((2a^3b^2c)^-3)/((3a^2b^3c)^2)

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