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

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

Step 1

Cancel the common factor of $a$ and $a^{2}$ .

Factor $a$ out of $2 a b^{2} c^{3}$ .

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

Cancel the common factors.

Factor $a$ out of $3 a^{2} c$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Cancel the common factor of $c^{3}$ and $c$ .

Factor $c$ out of $2 b^{2} c^{3}$ .

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

Cancel the common factors.

Factor $c$ out of $3 a c$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 4

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

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

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

Apply the product rule to $3 a$ .

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

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

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

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

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

Step 5

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

$\frac{27 a^{3}}{2^{3} {(b^{2})}^{3} {(c^{2})}^{3}}$

Step 6

Simplify the denominator.

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

$\frac{27 a^{3}}{8 {(b^{2})}^{3} {(c^{2})}^{3}}$

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

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

$\frac{27 a^{3}}{8 b^{2 \cdot 3} {(c^{2})}^{3}}$

Multiply $2$ by $3$ .

$\frac{27 a^{3}}{8 b^{6} {(c^{2})}^{3}}$

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

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

$\frac{27 a^{3}}{8 b^{6} c^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{27 a^{3}}{8 b^{6} c^{6}}$

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

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