Pre-Algebra Examples

$\frac{\sqrt{4 a c}}{\sqrt[3]{a^{2} c}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4 a c$ as $2^{2} (a c)$ .

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Step 1.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.1.2

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Multiply $2$ by $2$ .

$\frac{2 \sqrt{a c} \sqrt[3]{a^{4} c^{2}}}{a^{2} c}$

Step 4.4

Rewrite $a^{4} c^{2}$ as $a^{3} (a c^{2})$ .

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Step 4.4.1

Factor out $a^{3}$ .

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

Step 4.4.2

Add parentheses.

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

Step 4.5

Pull terms out from under the radical.

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

Step 4.6

Combine exponents.

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Step 4.6.1

Rewrite the expression using the least common index of $6$ .

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Step 4.6.1.1

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

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

Step 4.6.1.2

Rewrite ${(a c)}^{\frac{1}{2}}$ as ${(a c)}^{\frac{3}{6}}$ .

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

Step 4.6.1.3

Rewrite ${(a c)}^{\frac{3}{6}}$ as $\sqrt[6]{{(a c)}^{3}}$ .

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

Step 4.6.1.4

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

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

Step 4.6.1.5

Rewrite ${(a c^{2})}^{\frac{1}{3}}$ as ${(a c^{2})}^{\frac{2}{6}}$ .

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

Step 4.6.1.6

Rewrite ${(a c^{2})}^{\frac{2}{6}}$ as $\sqrt[6]{{(a c^{2})}^{2}}$ .

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

Step 4.6.2

Combine using the product rule for radicals.

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

Step 4.6.3

Apply the product rule to $a c$ .

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

Step 4.6.4

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

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

Step 4.6.5

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

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Step 4.6.5.1

Move $a^{2}$ .

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

Step 4.6.5.2

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

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

Step 4.6.5.3

Add $2$ and $3$ .

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

Step 4.6.6

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

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Step 4.6.6.1

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

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

Step 4.6.6.2

Multiply $2$ by $2$ .

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

Step 4.6.7

Multiply $c^{3}$ by $c^{4}$ by adding the exponents.

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Step 4.6.7.1

Move $c^{4}$ .

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

Step 4.6.7.2

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

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

Step 4.6.7.3

Add $4$ and $3$ .

$\frac{2 a \sqrt[6]{a^{5} c^{7}}}{a^{2} c}$

Step 4.7

Rewrite $a^{5} c^{7}$ as $c^{6} (a^{5} c)$ .

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Step 4.7.1

Factor out $c^{6}$ .

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

Step 4.7.2

Reorder $a^{5}$ and $c^{6}$ .

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

Step 4.7.3

Add parentheses.

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

Step 4.8

Pull terms out from under the radical.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Multiply $2$ by $2$ .

$\frac{2 \sqrt{a c} \sqrt[3]{a^{4} c^{2}}}{a^{2} c}$

Step 5.5

Rewrite $a^{4} c^{2}$ as $a^{3} (a c^{2})$ .

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Step 5.5.1

Factor out $a^{3}$ .

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

Step 5.5.2

Add parentheses.

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

Step 5.6

Pull terms out from under the radical.

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

Step 5.7

Combine exponents.

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Step 5.7.1

Rewrite the expression using the least common index of $6$ .

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Step 5.7.1.1

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

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

Step 5.7.1.2

Rewrite ${(a c)}^{\frac{1}{2}}$ as ${(a c)}^{\frac{3}{6}}$ .

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

Step 5.7.1.3

Rewrite ${(a c)}^{\frac{3}{6}}$ as $\sqrt[6]{{(a c)}^{3}}$ .

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

Step 5.7.1.4

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

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

Step 5.7.1.5

Rewrite ${(a c^{2})}^{\frac{1}{3}}$ as ${(a c^{2})}^{\frac{2}{6}}$ .

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

Step 5.7.1.6

Rewrite ${(a c^{2})}^{\frac{2}{6}}$ as $\sqrt[6]{{(a c^{2})}^{2}}$ .

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

Step 5.7.2

Combine using the product rule for radicals.

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

Step 5.7.3

Apply the product rule to $a c$ .

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

Step 5.7.4

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

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

Step 5.7.5

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

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Step 5.7.5.1

Move $a^{2}$ .

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

Step 5.7.5.2

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

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

Step 5.7.5.3

Add $2$ and $3$ .

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

Step 5.7.6

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

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Step 5.7.6.1

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

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

Step 5.7.6.2

Multiply $2$ by $2$ .

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

Step 5.7.7

Multiply $c^{3}$ by $c^{4}$ by adding the exponents.

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Step 5.7.7.1

Move $c^{4}$ .

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

Step 5.7.7.2

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

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

Step 5.7.7.3

Add $4$ and $3$ .

$\frac{2 a \sqrt[6]{a^{5} c^{7}}}{a^{2} c}$

Step 5.8

Rewrite $a^{5} c^{7}$ as $c^{6} (a^{5} c)$ .

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Step 5.8.1

Factor out $c^{6}$ .

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

Step 5.8.2

Reorder $a^{5}$ and $c^{6}$ .

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

Step 5.8.3

Add parentheses.

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

Step 5.9

Pull terms out from under the radical.

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

Step 5.10

Remove unnecessary parentheses.

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

Step 6

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

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Step 6.1

Factor $a$ out of $2 a c \sqrt[6]{a^{5} c}$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

Cancel the common factor of $c$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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