Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{2}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{2}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.2.2

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 1.2.3

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Simplify.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Combine the numerators over the common denominator.

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

Step 3.1.4

Add $2$ and $1$ .

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

Step 3.1.5

Cancel the common factor of $3$ .

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Step 3.1.5.1

Cancel the common factor.

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

Step 3.1.5.2

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 4

Simplify each term.

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Step 4.1

Simplify.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

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

Step 4.2.2.4

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

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

Step 4.2.2.5

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

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

Step 4.2.2.6

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

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

Step 4.2.3

Combine using the product rule for radicals.

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

Step 4.2.4

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

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Step 4.2.4.1

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

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

Step 4.2.4.2

Add $4$ and $3$ .

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Factor out $x^{6}$ .

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

Step 4.3.2

Pull terms out from under the radical.

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

Step 4.4

Cancel the common factor of $x$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

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

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

Step 4.5

Multiply $2$ by $- 1$ .

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