Simplify ((24a^11b^16c^6)/(18a^6b^6c^6))^3

${(\frac{24 a^{11} b^{16} c^{6}}{18 a^{6} b^{6} c^{6}})}^{3}$

Step 1

Cancel the common factor of $24$ and $18$ .

Factor $6$ out of $24 a^{11} b^{16} c^{6}$ .

${(\frac{6 (4 a^{11} b^{16} c^{6})}{18 a^{6} b^{6} c^{6}})}^{3}$

Cancel the common factors.

Factor $6$ out of $18 a^{6} b^{6} c^{6}$ .

${(\frac{6 (4 a^{11} b^{16} c^{6})}{6 (3 a^{6} b^{6} c^{6})})}^{3}$

Cancel the common factor.

${(\frac{6 (4 a^{11} b^{16} c^{6})}{6 (3 a^{6} b^{6} c^{6})})}^{3}$

Rewrite the expression.

${(\frac{4 a^{11} b^{16} c^{6}}{3 a^{6} b^{6} c^{6}})}^{3}$

Step 2

Cancel the common factor of $a^{11}$ and $a^{6}$ .

Factor $a^{6}$ out of $4 a^{11} b^{16} c^{6}$ .

${(\frac{a^{6} (4 a^{5} b^{16} c^{6})}{3 a^{6} b^{6} c^{6}})}^{3}$

Cancel the common factors.

Factor $a^{6}$ out of $3 a^{6} b^{6} c^{6}$ .

${(\frac{a^{6} (4 a^{5} b^{16} c^{6})}{a^{6} (3 b^{6} c^{6})})}^{3}$

Cancel the common factor.

${(\frac{a^{6} (4 a^{5} b^{16} c^{6})}{a^{6} (3 b^{6} c^{6})})}^{3}$

Rewrite the expression.

${(\frac{4 a^{5} b^{16} c^{6}}{3 b^{6} c^{6}})}^{3}$

Step 3

Cancel the common factor of $b^{16}$ and $b^{6}$ .

Factor $b^{6}$ out of $4 a^{5} b^{16} c^{6}$ .

${(\frac{b^{6} (4 a^{5} b^{10} c^{6})}{3 b^{6} c^{6}})}^{3}$

Cancel the common factors.

Factor $b^{6}$ out of $3 b^{6} c^{6}$ .

${(\frac{b^{6} (4 a^{5} b^{10} c^{6})}{b^{6} (3 c^{6})})}^{3}$

Cancel the common factor.

${(\frac{b^{6} (4 a^{5} b^{10} c^{6})}{b^{6} (3 c^{6})})}^{3}$

Rewrite the expression.

${(\frac{4 a^{5} b^{10} c^{6}}{3 c^{6}})}^{3}$

Step 4

Cancel the common factor of $c^{6}$ .

Cancel the common factor.

${(\frac{4 a^{5} b^{10} c^{6}}{3 c^{6}})}^{3}$

Rewrite the expression.

${(\frac{4 a^{5} b^{10}}{3})}^{3}$

Step 5

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

Apply the product rule to $\frac{4 a^{5} b^{10}}{3}$ .

$\frac{{(4 a^{5} b^{10})}^{3}}{3^{3}}$

Apply the product rule to $4 a^{5} b^{10}$ .

$\frac{{(4 a^{5})}^{3} {(b^{10})}^{3}}{3^{3}}$

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

$\frac{4^{3} {(a^{5})}^{3} {(b^{10})}^{3}}{3^{3}}$

Step 6

Simplify the numerator.

Raise $4$ to the power of $3$ .

$\frac{64 {(a^{5})}^{3} {(b^{10})}^{3}}{3^{3}}$

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

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

$\frac{64 a^{5 \cdot 3} {(b^{10})}^{3}}{3^{3}}$

Multiply $5$ by $3$ .

$\frac{64 a^{15} {(b^{10})}^{3}}{3^{3}}$

Multiply the exponents in ${(b^{10})}^{3}$ .

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

$\frac{64 a^{15} b^{10 \cdot 3}}{3^{3}}$

Multiply $10$ by $3$ .

$\frac{64 a^{15} b^{30}}{3^{3}}$

Step 7

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

$\frac{64 a^{15} b^{30}}{27}$

Simplify ((24a^11b^16c^6)/(18a^6b^6c^6))^3

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