Simplify ( cube root of 12ab^3c^2)/( cube root of 10a^3bc)

$\frac{\sqrt[3]{12 a b^{3} c^{2}}}{\sqrt[3]{10 a^{3} b c}}$

Step 1

Combine $\sqrt[3]{12 a b^{3} c^{2}}$ and $\sqrt[3]{10 a^{3} b c}$ into a single radical.

$\sqrt[3]{\frac{12 a b^{3} c^{2}}{10 a^{3} b c}}$

Step 2

Reduce the expression $\frac{12 a b^{3} c^{2}}{10 a^{3} b c}$ by cancelling the common factors.

Factor $2$ out of $12 a b^{3} c^{2}$ .

$\sqrt[3]{\frac{2 (6 a b^{3} c^{2})}{10 a^{3} b c}}$

Factor $2$ out of $10 a^{3} b c$ .

$\sqrt[3]{\frac{2 (6 a b^{3} c^{2})}{2 (5 a^{3} b c)}}$

Cancel the common factor.

$\sqrt[3]{\frac{2 (6 a b^{3} c^{2})}{2 (5 a^{3} b c)}}$

Rewrite the expression.

$\sqrt[3]{\frac{6 a b^{3} c^{2}}{5 a^{3} b c}}$

Step 3

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

Factor $a$ out of $6 a b^{3} c^{2}$ .

$\sqrt[3]{\frac{a (6 b^{3} c^{2})}{5 a^{3} b c}}$

Cancel the common factors.

Factor $a$ out of $5 a^{3} b c$ .

$\sqrt[3]{\frac{a (6 b^{3} c^{2})}{a (5 a^{2} b c)}}$

Cancel the common factor.

$\sqrt[3]{\frac{a (6 b^{3} c^{2})}{a (5 a^{2} b c)}}$

Rewrite the expression.

$\sqrt[3]{\frac{6 b^{3} c^{2}}{5 a^{2} b c}}$

Step 4

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

Factor $b$ out of $6 b^{3} c^{2}$ .

$\sqrt[3]{\frac{b (6 b^{2} c^{2})}{5 a^{2} b c}}$

Cancel the common factors.

Factor $b$ out of $5 a^{2} b c$ .

$\sqrt[3]{\frac{b (6 b^{2} c^{2})}{b (5 a^{2} c)}}$

Cancel the common factor.

$\sqrt[3]{\frac{b (6 b^{2} c^{2})}{b (5 a^{2} c)}}$

Rewrite the expression.

$\sqrt[3]{\frac{6 b^{2} c^{2}}{5 a^{2} c}}$

Step 5

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

Factor $c$ out of $6 b^{2} c^{2}$ .

$\sqrt[3]{\frac{c (6 b^{2} c)}{5 a^{2} c}}$

Cancel the common factors.

Factor $c$ out of $5 a^{2} c$ .

$\sqrt[3]{\frac{c (6 b^{2} c)}{c (5 a^{2})}}$

Cancel the common factor.

$\sqrt[3]{\frac{c (6 b^{2} c)}{c (5 a^{2})}}$

Rewrite the expression.

$\sqrt[3]{\frac{6 b^{2} c}{5 a^{2}}}$

Step 6

Rewrite $\sqrt[3]{\frac{6 b^{2} c}{5 a^{2}}}$ as $\frac{\sqrt[3]{6 b^{2} c}}{\sqrt[3]{5 a^{2}}}$ .

$\frac{\sqrt[3]{6 b^{2} c}}{\sqrt[3]{5 a^{2}}}$

Step 7

Multiply $\frac{\sqrt[3]{6 b^{2} c}}{\sqrt[3]{5 a^{2}}}$ by $\frac{{\sqrt[3]{5 a^{2}}}^{2}}{{\sqrt[3]{5 a^{2}}}^{2}}$ .

$\frac{\sqrt[3]{6 b^{2} c}}{\sqrt[3]{5 a^{2}}} \cdot \frac{{\sqrt[3]{5 a^{2}}}^{2}}{{\sqrt[3]{5 a^{2}}}^{2}}$

Step 8

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{6 b^{2} c}}{\sqrt[3]{5 a^{2}}}$ by $\frac{{\sqrt[3]{5 a^{2}}}^{2}}{{\sqrt[3]{5 a^{2}}}^{2}}$ .

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{\sqrt[3]{5 a^{2}} {\sqrt[3]{5 a^{2}}}^{2}}$

Raise $\sqrt[3]{5 a^{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{\sqrt[3]{5 a^{2}}}^{1} {\sqrt[3]{5 a^{2}}}^{2}}$

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

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{\sqrt[3]{5 a^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{\sqrt[3]{5 a^{2}}}^{3}}$

Rewrite ${\sqrt[3]{5 a^{2}}}^{3}$ as $5 a^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 a^{2}}$ as ${(5 a^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{({(5 a^{2})}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{(5 a^{2})}^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{(5 a^{2})}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{(5 a^{2})}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{{(5 a^{2})}^{1}}$

Simplify.

$\frac{\sqrt[3]{6 b^{2} c} {\sqrt[3]{5 a^{2}}}^{2}}{5 a^{2}}$

Step 9

Simplify the numerator.

Rewrite ${\sqrt[3]{5 a^{2}}}^{2}$ as $\sqrt[3]{{(5 a^{2})}^{2}}$ .

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{{(5 a^{2})}^{2}}}{5 a^{2}}$

Apply the product rule to $5 a^{2}$ .

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{5^{2} {(a^{2})}^{2}}}{5 a^{2}}$

Raise $5$ to the power of $2$ .

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{25 {(a^{2})}^{2}}}{5 a^{2}}$

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

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

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{25 a^{2 \cdot 2}}}{5 a^{2}}$

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{25 a^{4}}}{5 a^{2}}$

Rewrite $25 a^{4}$ as $a^{3} \cdot (25 a)$ .

Factor out $a^{3}$ .

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{25 (a^{3} a)}}{5 a^{2}}$

Reorder $25$ and $a^{3}$ .

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{a^{3} \cdot 25 a}}{5 a^{2}}$

Add parentheses.

$\frac{\sqrt[3]{6 b^{2} c} \sqrt[3]{a^{3} \cdot (25 a)}}{5 a^{2}}$

Pull terms out from under the radical.

$\frac{\sqrt[3]{6 b^{2} c} a \sqrt[3]{25 a}}{5 a^{2}}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{a \sqrt[3]{6 b^{2} c (25 a)}}{5 a^{2}}$

Multiply $25$ by $6$ .

$\frac{a \sqrt[3]{150 b^{2} c a}}{5 a^{2}}$

Step 10

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

Factor $a$ out of $a \sqrt[3]{150 b^{2} c a}$ .

$\frac{a (\sqrt[3]{150 b^{2} c a})}{5 a^{2}}$

Cancel the common factors.

Factor $a$ out of $5 a^{2}$ .

$\frac{a (\sqrt[3]{150 b^{2} c a})}{a (5 a)}$

Cancel the common factor.

$\frac{a \sqrt[3]{150 b^{2} c a}}{a (5 a)}$

Rewrite the expression.

$\frac{\sqrt[3]{150 b^{2} c a}}{5 a}$

Simplify ( cube root of 12ab^3c^2)/( cube root of 10a^3bc)

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