Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Combine and simplify the denominator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Add $1$ and $1$ .

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

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

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

Step 1.1.2.6.2

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

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

Step 1.1.2.6.3

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

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

Step 1.1.2.6.4

Cancel the common factor of $2$ .

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Step 1.1.2.6.4.1

Cancel the common factor.

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

Step 1.1.2.6.4.2

Rewrite the expression.

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

Step 1.1.2.6.5

Simplify.

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

Step 1.2

Rewrite ${(\sqrt{x} - \frac{\sqrt{x}}{x})}^{2}$ as $(\sqrt{x} - \frac{\sqrt{x}}{x}) (\sqrt{x} - \frac{\sqrt{x}}{x})$ .

$(\sqrt{x} - \frac{\sqrt{x}}{x}) (\sqrt{x} - \frac{\sqrt{x}}{x})$

Step 2

Expand $(\sqrt{x} - \frac{\sqrt{x}}{x}) (\sqrt{x} - \frac{\sqrt{x}}{x})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{x} (\sqrt{x} - \frac{\sqrt{x}}{x}) - \frac{\sqrt{x}}{x} (\sqrt{x} - \frac{\sqrt{x}}{x})$

Step 2.2

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} (- \frac{\sqrt{x}}{x}) - \frac{\sqrt{x}}{x} (\sqrt{x} - \frac{\sqrt{x}}{x})$

Step 2.3

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} (- \frac{\sqrt{x}}{x}) - \frac{\sqrt{x}}{x} \sqrt{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

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

Step 3.1.2.4.2

Rewrite the expression.

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

Step 3.1.2.5

Simplify.

$x + \sqrt{x} (- \frac{\sqrt{x}}{x}) - \frac{\sqrt{x}}{x} \sqrt{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.3

Rewrite using the commutative property of multiplication.

$x - \sqrt{x} \frac{\sqrt{x}}{x} - \frac{\sqrt{x}}{x} \sqrt{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.4

Multiply $- \sqrt{x} \frac{\sqrt{x}}{x}$ .

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Step 3.1.4.1

Combine $\frac{\sqrt{x}}{x}$ and $\sqrt{x}$ .

$x - \frac{\sqrt{x} \sqrt{x}}{x} - \frac{\sqrt{x}}{x} \sqrt{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

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

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

Step 3.1.4.5

Add $1$ and $1$ .

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

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

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

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

Step 3.1.5.4.2

Rewrite the expression.

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

Step 3.1.5.5

Simplify.

$x - \frac{x}{x} - \frac{\sqrt{x}}{x} \sqrt{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.6

Cancel the common factor of $x$ .

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Step 3.1.6.1

Cancel the common factor.

$x - \frac{x}{x} - \frac{\sqrt{x}}{x} \sqrt{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.6.2

Rewrite the expression.

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

Step 3.1.7

Multiply $- 1$ by $1$ .

$x - 1 - \frac{\sqrt{x}}{x} \sqrt{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.8

Multiply $- \frac{\sqrt{x}}{x} \sqrt{x}$ .

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Step 3.1.8.1

Combine $\sqrt{x}$ and $\frac{\sqrt{x}}{x}$ .

$x - 1 - \frac{\sqrt{x} \sqrt{x}}{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.8.2

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

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

Step 3.1.8.3

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

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

Step 3.1.8.4

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

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

Step 3.1.8.5

Add $1$ and $1$ .

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

Step 3.1.9

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

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Step 3.1.9.1

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

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

Step 3.1.9.2

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

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

Step 3.1.9.3

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

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

Step 3.1.9.4

Cancel the common factor of $2$ .

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Step 3.1.9.4.1

Cancel the common factor.

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

Step 3.1.9.4.2

Rewrite the expression.

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

Step 3.1.9.5

Simplify.

$x - 1 - \frac{x}{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.10

Cancel the common factor of $x$ .

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Step 3.1.10.1

Cancel the common factor.

$x - 1 - \frac{x}{x} - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.10.2

Rewrite the expression.

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

Step 3.1.11

Multiply $- 1$ by $1$ .

$x - 1 - 1 - \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$

Step 3.1.12

Multiply $- \frac{\sqrt{x}}{x} (- \frac{\sqrt{x}}{x})$ .

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Step 3.1.12.1

Multiply $- 1$ by $- 1$ .

$x - 1 - 1 + 1 \frac{\sqrt{x}}{x} \frac{\sqrt{x}}{x}$

Step 3.1.12.2

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

$x - 1 - 1 + \frac{\sqrt{x}}{x} \cdot \frac{\sqrt{x}}{x}$

Step 3.1.12.3

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

$x - 1 - 1 + \frac{\sqrt{x} \sqrt{x}}{x \cdot x}$

Step 3.1.12.4

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

$x - 1 - 1 + \frac{{\sqrt{x}}^{1} \sqrt{x}}{x \cdot x}$

Step 3.1.12.5

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

$x - 1 - 1 + \frac{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}{x \cdot x}$

Step 3.1.12.6

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

$x - 1 - 1 + \frac{{\sqrt{x}}^{1 + 1}}{x \cdot x}$

Step 3.1.12.7

Add $1$ and $1$ .

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

Step 3.1.12.8

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

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

Step 3.1.12.9

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

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

Step 3.1.12.10

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

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

Step 3.1.12.11

Add $1$ and $1$ .

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

Step 3.1.13

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

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Step 3.1.13.1

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

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

Step 3.1.13.2

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

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

Step 3.1.13.3

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

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

Step 3.1.13.4

Cancel the common factor of $2$ .

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Step 3.1.13.4.1

Cancel the common factor.

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

Step 3.1.13.4.2

Rewrite the expression.

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

Step 3.1.13.5

Simplify.

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

Step 3.1.14

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

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Step 3.1.14.1

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

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

Step 3.1.14.2

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

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

Step 3.1.14.3

Cancel the common factors.

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Step 3.1.14.3.1

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

$x - 1 - 1 + \frac{x \cdot 1}{x \cdot x}$

Step 3.1.14.3.2

Cancel the common factor.

$x - 1 - 1 + \frac{x \cdot 1}{x \cdot x}$

Step 3.1.14.3.3

Rewrite the expression.

$x - 1 - 1 + \frac{1}{x}$

Step 3.2

Subtract $1$ from $- 1$ .

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