Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

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 $x^{\frac{1}{2}}$ .

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

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.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Simplify.

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

Step 2

Apply the distributive property.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add $3$ and $1$ .

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

Step 3.5

Cancel the common factor of $4$ and $2$ .

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Step 3.5.1

Factor $2$ out of $4$ .

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

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $2$ out of $2$ .

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

Step 3.5.2.2

Cancel the common factor.

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

Step 3.5.2.3

Rewrite the expression.

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

Step 3.5.2.4

Divide $2$ by $1$ .

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

Step 4

Rewrite using the commutative property of multiplication.

$x^{2} - x^{\frac{3}{2}} \frac{\sqrt{x}}{x}$

Step 5

Simplify each term.

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Step 5.1

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

$x^{2} - \frac{\sqrt{x} x^{\frac{3}{2}}}{x}$

Step 5.2

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

$x^{2} - \frac{x^{\frac{1}{2}} x^{\frac{3}{2}}}{x}$

Step 5.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

Add $1$ and $3$ .

$x^{2} - \frac{x^{\frac{4}{2}}}{x}$

Step 5.6

Cancel the common factor of $4$ and $2$ .

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Step 5.6.1

Factor $2$ out of $4$ .

$x^{2} - \frac{x^{\frac{2 \cdot 2}{2}}}{x}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $2$ out of $2$ .

$x^{2} - \frac{x^{\frac{2 \cdot 2}{2 (1)}}}{x}$

Step 5.6.2.2

Cancel the common factor.

$x^{2} - \frac{x^{\frac{2 \cdot 2}{2 \cdot 1}}}{x}$

Step 5.6.2.3

Rewrite the expression.

$x^{2} - \frac{x^{\frac{2}{1}}}{x}$

Step 5.6.2.4

Divide $2$ by $1$ .

$x^{2} - \frac{x^{2}}{x}$

Step 5.7

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

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Step 5.7.1

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

$x^{2} - \frac{x \cdot x}{x}$

Step 5.7.2

Cancel the common factors.

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Step 5.7.2.1

Raise $x$ to the power of $1$ .

$x^{2} - \frac{x \cdot x}{x^{1}}$

Step 5.7.2.2

Factor $x$ out of $x^{1}$ .

$x^{2} - \frac{x \cdot x}{x \cdot 1}$

Step 5.7.2.3

Cancel the common factor.

$x^{2} - \frac{x \cdot x}{x \cdot 1}$

Step 5.7.2.4

Rewrite the expression.

$x^{2} - \frac{x}{1}$

Step 5.7.2.5

Divide $x$ by $1$ .

$x^{2} - x$