Algebra Examples

$\frac{\sqrt{8 x}}{\sqrt[3]{16 x^{2}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8 x$ as $2^{2} \cdot (2 x)$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2) x}}{\sqrt[3]{16 x^{2}}}$

Step 1.1.2

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

$\frac{\sqrt{2^{2} \cdot 2 x}}{\sqrt[3]{16 x^{2}}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt{2^{2} \cdot (2 x)}}{\sqrt[3]{16 x^{2}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2 x}}{\sqrt[3]{16 x^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16 x^{2}$ as $2^{3} \cdot (2 x^{2})$ .

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Step 2.1.1

Factor $8$ out of $16$ .

$\frac{2 \sqrt{2 x}}{\sqrt[3]{8 (2) x^{2}}}$

Step 2.1.2

Rewrite $8$ as $2^{3}$ .

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

Step 2.1.3

Add parentheses.

$\frac{2 \sqrt{2 x}}{\sqrt[3]{2^{3} \cdot (2 x^{2})}}$

Step 2.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2 x}}{2 \sqrt[3]{2 x^{2}}}$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

$\frac{2 \sqrt{2 x}}{2 \sqrt[3]{2 x^{2}}}$

Step 3.2

Rewrite the expression.

$\frac{\sqrt{2 x}}{\sqrt[3]{2 x^{2}}}$

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

$\frac{\sqrt{2 x} {\sqrt[3]{2 x^{2}}}^{2}}{\sqrt[3]{2 x^{2}} {\sqrt[3]{2 x^{2}}}^{2}}$

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $2$ .

$\frac{\sqrt{2 x} {\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{3}}$

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

$\frac{\sqrt{2 x} {\sqrt[3]{2 x^{2}}}^{2}}{2 x^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

$\frac{\sqrt{2 x} \sqrt[3]{4 {(x^{2})}^{2}}}{2 x^{2}}$

Step 6.4

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

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Step 6.4.1

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

$\frac{\sqrt{2 x} \sqrt[3]{4 x^{2 \cdot 2}}}{2 x^{2}}$

Step 6.4.2

Multiply $2$ by $2$ .

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

Step 6.5

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

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Step 6.5.1

Factor out $x^{3}$ .

$\frac{\sqrt{2 x} \sqrt[3]{4 (x^{3} x)}}{2 x^{2}}$

Step 6.5.2

Reorder $4$ and $x^{3}$ .

$\frac{\sqrt{2 x} \sqrt[3]{x^{3} \cdot 4 x}}{2 x^{2}}$

Step 6.5.3

Add parentheses.

$\frac{\sqrt{2 x} \sqrt[3]{x^{3} \cdot (4 x)}}{2 x^{2}}$

Step 6.6

Pull terms out from under the radical.

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

Step 6.7

Combine exponents.

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Step 6.7.1

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

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Step 6.7.1.1

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

$\frac{x ({(2 x)}^{\frac{1}{2}} \sqrt[3]{4 x})}{2 x^{2}}$

Step 6.7.1.2

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

$\frac{x ({(2 x)}^{\frac{3}{6}} \sqrt[3]{4 x})}{2 x^{2}}$

Step 6.7.1.3

Rewrite ${(2 x)}^{\frac{3}{6}}$ as $\sqrt[6]{{(2 x)}^{3}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} \sqrt[3]{4 x})}{2 x^{2}}$

Step 6.7.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x}$ as ${(4 x)}^{\frac{1}{3}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} {(4 x)}^{\frac{1}{3}})}{2 x^{2}}$

Step 6.7.1.5

Rewrite ${(4 x)}^{\frac{1}{3}}$ as ${(4 x)}^{\frac{2}{6}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} {(4 x)}^{\frac{2}{6}})}{2 x^{2}}$

Step 6.7.1.6

Rewrite ${(4 x)}^{\frac{2}{6}}$ as $\sqrt[6]{{(4 x)}^{2}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} \sqrt[6]{{(4 x)}^{2}})}{2 x^{2}}$

Step 6.7.2

Combine using the product rule for radicals.

$\frac{x \sqrt[6]{{(2 x)}^{3} {(4 x)}^{2}}}{2 x^{2}}$

Step 6.7.3

Apply the product rule to $2 x$ .

$\frac{x \sqrt[6]{2^{3} x^{3} {(4 x)}^{2}}}{2 x^{2}}$

Step 6.7.4

Apply the product rule to $4 x$ .

$\frac{x \sqrt[6]{2^{3} x^{3} \cdot 4^{2} x^{2}}}{2 x^{2}}$

Step 6.7.5

Combine exponents.

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Step 6.7.5.1

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

$\frac{x \sqrt[6]{{(2^{2})}^{2} \cdot 2^{3} x^{3} x^{2}}}{2 x^{2}}$

Step 6.7.5.2

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

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Step 6.7.5.2.1

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

$\frac{x \sqrt[6]{2^{2 \cdot 2} \cdot 2^{3} x^{3} x^{2}}}{2 x^{2}}$

Step 6.7.5.2.2

Multiply $2$ by $2$ .

$\frac{x \sqrt[6]{2^{4} \cdot 2^{3} x^{3} x^{2}}}{2 x^{2}}$

Step 6.7.5.3

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

$\frac{x \sqrt[6]{2^{4 + 3} x^{3} x^{2}}}{2 x^{2}}$

Step 6.7.5.4

Add $4$ and $3$ .

$\frac{x \sqrt[6]{2^{7} x^{3} x^{2}}}{2 x^{2}}$

Step 6.7.5.5

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

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Step 6.7.5.5.1

Move $x^{2}$ .

$\frac{x \sqrt[6]{2^{7} (x^{2} x^{3})}}{2 x^{2}}$

Step 6.7.5.5.2

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

$\frac{x \sqrt[6]{2^{7} x^{2 + 3}}}{2 x^{2}}$

Step 6.7.5.5.3

Add $2$ and $3$ .

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

Step 6.7.6

Raise $2$ to the power of $7$ .

$\frac{x \sqrt[6]{128 x^{5}}}{2 x^{2}}$

Step 6.8

Rewrite $128 x^{5}$ as $2^{6} \cdot (2 x^{5})$ .

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Step 6.8.1

Factor $64$ out of $128$ .

$\frac{x \sqrt[6]{64 (2) x^{5}}}{2 x^{2}}$

Step 6.8.2

Rewrite $64$ as $2^{6}$ .

$\frac{x \sqrt[6]{2^{6} \cdot 2 x^{5}}}{2 x^{2}}$

Step 6.8.3

Add parentheses.

$\frac{x \sqrt[6]{2^{6} \cdot (2 x^{5})}}{2 x^{2}}$

Step 6.9

Pull terms out from under the radical.

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

Step 7

Simplify the numerator.

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Step 7.1

Rewrite.

$\frac{\sqrt{2 x} {\sqrt[3]{2 x^{2}}}^{2}}{2 x^{2}}$

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

$\frac{\sqrt{2 x} \sqrt[3]{4 {(x^{2})}^{2}}}{2 x^{2}}$

Step 7.5

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

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Step 7.5.1

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

$\frac{\sqrt{2 x} \sqrt[3]{4 x^{2 \cdot 2}}}{2 x^{2}}$

Step 7.5.2

Multiply $2$ by $2$ .

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

Step 7.6

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

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Step 7.6.1

Factor out $x^{3}$ .

$\frac{\sqrt{2 x} \sqrt[3]{4 (x^{3} x)}}{2 x^{2}}$

Step 7.6.2

Reorder $4$ and $x^{3}$ .

$\frac{\sqrt{2 x} \sqrt[3]{x^{3} \cdot 4 x}}{2 x^{2}}$

Step 7.6.3

Add parentheses.

$\frac{\sqrt{2 x} \sqrt[3]{x^{3} \cdot (4 x)}}{2 x^{2}}$

Step 7.7

Pull terms out from under the radical.

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

Step 7.8

Combine exponents.

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Step 7.8.1

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

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Step 7.8.1.1

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

$\frac{x ({(2 x)}^{\frac{1}{2}} \sqrt[3]{4 x})}{2 x^{2}}$

Step 7.8.1.2

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

$\frac{x ({(2 x)}^{\frac{3}{6}} \sqrt[3]{4 x})}{2 x^{2}}$

Step 7.8.1.3

Rewrite ${(2 x)}^{\frac{3}{6}}$ as $\sqrt[6]{{(2 x)}^{3}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} \sqrt[3]{4 x})}{2 x^{2}}$

Step 7.8.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x}$ as ${(4 x)}^{\frac{1}{3}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} {(4 x)}^{\frac{1}{3}})}{2 x^{2}}$

Step 7.8.1.5

Rewrite ${(4 x)}^{\frac{1}{3}}$ as ${(4 x)}^{\frac{2}{6}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} {(4 x)}^{\frac{2}{6}})}{2 x^{2}}$

Step 7.8.1.6

Rewrite ${(4 x)}^{\frac{2}{6}}$ as $\sqrt[6]{{(4 x)}^{2}}$ .

$\frac{x (\sqrt[6]{{(2 x)}^{3}} \sqrt[6]{{(4 x)}^{2}})}{2 x^{2}}$

Step 7.8.2

Combine using the product rule for radicals.

$\frac{x \sqrt[6]{{(2 x)}^{3} {(4 x)}^{2}}}{2 x^{2}}$

Step 7.8.3

Apply the product rule to $2 x$ .

$\frac{x \sqrt[6]{2^{3} x^{3} {(4 x)}^{2}}}{2 x^{2}}$

Step 7.8.4

Apply the product rule to $4 x$ .

$\frac{x \sqrt[6]{2^{3} x^{3} \cdot 4^{2} x^{2}}}{2 x^{2}}$

Step 7.8.5

Combine exponents.

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Step 7.8.5.1

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

$\frac{x \sqrt[6]{{(2^{2})}^{2} \cdot 2^{3} x^{3} x^{2}}}{2 x^{2}}$

Step 7.8.5.2

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

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Step 7.8.5.2.1

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

$\frac{x \sqrt[6]{2^{2 \cdot 2} \cdot 2^{3} x^{3} x^{2}}}{2 x^{2}}$

Step 7.8.5.2.2

Multiply $2$ by $2$ .

$\frac{x \sqrt[6]{2^{4} \cdot 2^{3} x^{3} x^{2}}}{2 x^{2}}$

Step 7.8.5.3

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

$\frac{x \sqrt[6]{2^{4 + 3} x^{3} x^{2}}}{2 x^{2}}$

Step 7.8.5.4

Add $4$ and $3$ .

$\frac{x \sqrt[6]{2^{7} x^{3} x^{2}}}{2 x^{2}}$

Step 7.8.5.5

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

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Step 7.8.5.5.1

Move $x^{2}$ .

$\frac{x \sqrt[6]{2^{7} (x^{2} x^{3})}}{2 x^{2}}$

Step 7.8.5.5.2

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

$\frac{x \sqrt[6]{2^{7} x^{2 + 3}}}{2 x^{2}}$

Step 7.8.5.5.3

Add $2$ and $3$ .

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

Step 7.8.6

Raise $2$ to the power of $7$ .

$\frac{x \sqrt[6]{128 x^{5}}}{2 x^{2}}$

Step 7.9

Rewrite $128 x^{5}$ as $2^{6} \cdot (2 x^{5})$ .

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Step 7.9.1

Factor $64$ out of $128$ .

$\frac{x \sqrt[6]{64 (2) x^{5}}}{2 x^{2}}$

Step 7.9.2

Rewrite $64$ as $2^{6}$ .

$\frac{x \sqrt[6]{2^{6} \cdot 2 x^{5}}}{2 x^{2}}$

Step 7.9.3

Add parentheses.

$\frac{x \sqrt[6]{2^{6} \cdot (2 x^{5})}}{2 x^{2}}$

Step 7.10

Pull terms out from under the radical.

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

Step 7.11

Remove unnecessary parentheses.

$\frac{x \cdot 2 \sqrt[6]{2 x^{5}}}{2 x^{2}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

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

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Step 8.1.1

Factor $x$ out of $x \cdot 2 \sqrt[6]{2 x^{5}}$ .

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

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $x$ out of $2 x^{2}$ .

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

Step 8.1.2.2

Cancel the common factor.

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

Step 8.1.2.3

Rewrite the expression.

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

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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