Basic Math Examples

(\sqrt{x} - \sqrt{2}) (\sqrt{x} + \sqrt{2})

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

Step 1

Expand

(\sqrt{x} - \sqrt{2}) (\sqrt{x} + \sqrt{2})

$(\sqrt{x} - \sqrt{2}) (\sqrt{x} + \sqrt{2})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

\sqrt{x} (\sqrt{x} + \sqrt{2}) - \sqrt{2} (\sqrt{x} + \sqrt{2})

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

Step 1.2

Apply the distributive property.

\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} (\sqrt{x} + \sqrt{2})

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

Step 1.3

Apply the distributive property.

\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}

$\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}$

\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}

$\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in

\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}

$\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}$ .

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Step 2.1.1

Reorder the factors in the terms

\sqrt{x} \sqrt{2}

$\sqrt{x} \sqrt{2}$ and

- \sqrt{2} \sqrt{x}

$- \sqrt{2} \sqrt{x}$ .

\sqrt{x} \sqrt{x} + \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}

$\sqrt{x} \sqrt{x} + \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{x} - \sqrt{2} \sqrt{2}$

Step 2.1.2

Subtract

\sqrt{2} \sqrt{x}

$\sqrt{2} \sqrt{x}$ from

\sqrt{2} \sqrt{x}

$\sqrt{2} \sqrt{x}$ .

\sqrt{x} \sqrt{x} + 0 - \sqrt{2} \sqrt{2}

$\sqrt{x} \sqrt{x} + 0 - \sqrt{2} \sqrt{2}$

Step 2.1.3

Add

\sqrt{x} \sqrt{x}

$\sqrt{x} \sqrt{x}$ and

0

$0$ .

\sqrt{x} \sqrt{x} - \sqrt{2} \sqrt{2}

$\sqrt{x} \sqrt{x} - \sqrt{2} \sqrt{2}$

\sqrt{x} \sqrt{x} - \sqrt{2} \sqrt{2}

$\sqrt{x} \sqrt{x} - \sqrt{2} \sqrt{2}$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply

\sqrt{x} \sqrt{x}

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

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Step 2.2.1.1

Raise

\sqrt{x}

$\sqrt{x}$ to the power of

1

$1$ .

{\sqrt{x}}^{1} \sqrt{x} - \sqrt{2} \sqrt{2}

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

Step 2.2.1.2

Raise

\sqrt{x}

$\sqrt{x}$ to the power of

1

$1$ .

{\sqrt{x}}^{1} {\sqrt{x}}^{1} - \sqrt{2} \sqrt{2}

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

Step 2.2.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.2.1.4

Add

1

$1$ and

1

$1$ .

{\sqrt{x}}^{2} - \sqrt{2} \sqrt{2}

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

{\sqrt{x}}^{2} - \sqrt{2} \sqrt{2}

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

Step 2.2.2

Rewrite

{\sqrt{x}}^{2}

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

x

$x$ .

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Step 2.2.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

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

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

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

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

Step 2.2.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.2.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 2.2.2.4

Cancel the common factor of

2

$2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

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

Step 2.2.2.4.2

Rewrite the expression.

x^{1} - \sqrt{2} \sqrt{2}

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

x^{1} - \sqrt{2} \sqrt{2}

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

Step 2.2.2.5

Simplify.

x - \sqrt{2} \sqrt{2}

$x - \sqrt{2} \sqrt{2}$

x - \sqrt{2} \sqrt{2}

$x - \sqrt{2} \sqrt{2}$

Step 2.2.3

Multiply

- \sqrt{2} \sqrt{2}

$- \sqrt{2} \sqrt{2}$ .

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Step 2.2.3.1

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 2.2.3.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 2.2.3.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.2.3.4

Add

1

$1$ and

1

$1$ .

x - {\sqrt{2}}^{2}

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

x - {\sqrt{2}}^{2}

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

Step 2.2.4

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 2.2.4.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

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

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

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

Step 2.2.4.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.2.4.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 2.2.4.4

Cancel the common factor of

2

$2$ .

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Step 2.2.4.4.1

Cancel the common factor.

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

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

Step 2.2.4.4.2

Rewrite the expression.

x - 2^{1}

$x - 2^{1}$

x - 2^{1}

$x - 2^{1}$

Step 2.2.4.5

Evaluate the exponent.

x - 1 \cdot 2

$x - 1 \cdot 2$

x - 1 \cdot 2

$x - 1 \cdot 2$

Step 2.2.5

Multiply

- 1

$- 1$ by

2

$2$ .

x - 2

$x - 2$

x - 2

$x - 2$

x - 2

$x - 2$