Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

Apply the product rule to $2 x$ .

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

Step 3.3

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

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{2}}$ as ${(4 x^{2})}^{\frac{1}{3}}$ .

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Apply the product rule to $2 x$ .

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

Step 4.4

Apply the product rule to $4 x^{2}$ .

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

Step 4.5

Combine exponents.

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Step 4.5.1

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

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

Step 4.5.2

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

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Step 4.5.2.1

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

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

Step 4.5.2.2

Multiply $2$ by $2$ .

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

Step 4.5.3

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

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

Step 4.5.4

Add $4$ and $3$ .

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

Step 4.6

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

$\frac{\sqrt[6]{128 x^{3} {(x^{2})}^{2}}}{2 x}$

Step 4.7

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

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Step 4.7.1

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

$\frac{\sqrt[6]{128 x^{3} x^{2 \cdot 2}}}{2 x}$

Step 4.7.2

Multiply $2$ by $2$ .

$\frac{\sqrt[6]{128 x^{3} x^{4}}}{2 x}$

Step 4.8

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

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Step 4.8.1

Move $x^{4}$ .

$\frac{\sqrt[6]{128 (x^{4} x^{3})}}{2 x}$

Step 4.8.2

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

$\frac{\sqrt[6]{128 x^{4 + 3}}}{2 x}$

Step 4.8.3

Add $4$ and $3$ .

$\frac{\sqrt[6]{128 x^{7}}}{2 x}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $128 x^{7}$ as ${(2 x)}^{6} \cdot (2 x)$ .

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Step 5.1.1

Factor $64$ out of $128$ .

$\frac{\sqrt[6]{64 (2) x^{7}}}{2 x}$

Step 5.1.2

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

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

Step 5.1.3

Factor out $x^{6}$ .

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

Step 5.1.4

Move $2$ .

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

Step 5.1.5

Rewrite $2^{6} x^{6}$ as ${(2 x)}^{6}$ .

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

Step 5.1.6

Add parentheses.

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

Step 5.2

Pull terms out from under the radical.

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Divide $\sqrt[6]{2 x}$ by $1$ .

$\sqrt[6]{2 x}$