Simplify ((b^2c^2)/(3b^4))^3

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

Step 1

Cancel the common factor of $b^{2}$ and $b^{4}$ .

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

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

Cancel the common factors.

Factor $b^{2}$ out of $3 b^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

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

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

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

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

Step 3

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

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

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

Multiply $2$ by $3$ .

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

Step 4

Simplify the denominator.

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

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

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

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

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

Multiply $2$ by $3$ .

$\frac{c^{6}}{27 b^{6}}$