Basic Math Examples

$\frac{(5 + \sqrt{3}) (5 - \sqrt{3})}{\sqrt{22}}$

Step 1

Simplify the numerator.

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Step 1.1

Expand $(5 + \sqrt{3}) (5 - \sqrt{3})$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$\frac{5 (5 - \sqrt{3}) + \sqrt{3} (5 - \sqrt{3})}{\sqrt{22}}$

Step 1.1.2

Apply the distributive property.

$\frac{5 \cdot 5 + 5 (- \sqrt{3}) + \sqrt{3} (5 - \sqrt{3})}{\sqrt{22}}$

Step 1.1.3

Apply the distributive property.

$\frac{5 \cdot 5 + 5 (- \sqrt{3}) + \sqrt{3} \cdot 5 + \sqrt{3} (- \sqrt{3})}{\sqrt{22}}$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $5$ by $5$ .

$\frac{25 + 5 (- \sqrt{3}) + \sqrt{3} \cdot 5 + \sqrt{3} (- \sqrt{3})}{\sqrt{22}}$

Step 1.2.1.2

Multiply $- 1$ by $5$ .

$\frac{25 - 5 \sqrt{3} + \sqrt{3} \cdot 5 + \sqrt{3} (- \sqrt{3})}{\sqrt{22}}$

Step 1.2.1.3

Move $5$ to the left of $\sqrt{3}$ .

$\frac{25 - 5 \sqrt{3} + 5 \cdot \sqrt{3} + \sqrt{3} (- \sqrt{3})}{\sqrt{22}}$

Step 1.2.1.4

Multiply $\sqrt{3} (- \sqrt{3})$ .

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Step 1.2.1.4.1

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - ({\sqrt{3}}^{1} \sqrt{3})}{\sqrt{22}}$

Step 1.2.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{\sqrt{22}}$

Step 1.2.1.4.3

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

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - {\sqrt{3}}^{1 + 1}}{\sqrt{22}}$

Step 1.2.1.4.4

Add $1$ and $1$ .

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - {\sqrt{3}}^{2}}{\sqrt{22}}$

Step 1.2.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.2.1.5.1

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

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - {(3^{\frac{1}{2}})}^{2}}{\sqrt{22}}$

Step 1.2.1.5.2

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

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - 3^{\frac{1}{2} \cdot 2}}{\sqrt{22}}$

Step 1.2.1.5.3

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

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - 3^{\frac{2}{2}}}{\sqrt{22}}$

Step 1.2.1.5.4

Cancel the common factor of $2$ .

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Step 1.2.1.5.4.1

Cancel the common factor.

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - 3^{\frac{2}{2}}}{\sqrt{22}}$

Step 1.2.1.5.4.2

Rewrite the expression.

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - 3^{1}}{\sqrt{22}}$

Step 1.2.1.5.5

Evaluate the exponent.

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - 1 \cdot 3}{\sqrt{22}}$

Step 1.2.1.6

Multiply $- 1$ by $3$ .

$\frac{25 - 5 \sqrt{3} + 5 \sqrt{3} - 3}{\sqrt{22}}$

Step 1.2.2

Subtract $3$ from $25$ .

$\frac{22 - 5 \sqrt{3} + 5 \sqrt{3}}{\sqrt{22}}$

Step 1.2.3

Add $- 5 \sqrt{3}$ and $5 \sqrt{3}$ .

$\frac{22 + 0}{\sqrt{22}}$

Step 1.2.4

Add $22$ and $0$ .

$\frac{22}{\sqrt{22}}$

Step 2

Multiply $\frac{22}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$\frac{22}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{22}{\sqrt{22}}$ by $\frac{\sqrt{22}}{\sqrt{22}}$ .

$\frac{22 \sqrt{22}}{\sqrt{22} \sqrt{22}}$

Step 3.2

Raise $\sqrt{22}$ to the power of $1$ .

$\frac{22 \sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}}$

Step 3.3

Raise $\sqrt{22}$ to the power of $1$ .

$\frac{22 \sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}}$

Step 3.4

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

$\frac{22 \sqrt{22}}{{\sqrt{22}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{22 \sqrt{22}}{{\sqrt{22}}^{2}}$

Step 3.6

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$\frac{22 \sqrt{22}}{{(22^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{22 \sqrt{22}}{22^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{22 \sqrt{22}}{22^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{22 \sqrt{22}}{22^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{22 \sqrt{22}}{22^{1}}$

Step 3.6.5

Evaluate the exponent.

$\frac{22 \sqrt{22}}{22}$

Step 4

Cancel the common factor of $22$ .

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Step 4.1

Cancel the common factor.

$\frac{22 \sqrt{22}}{22}$

Step 4.2

Divide $\sqrt{22}$ by $1$ .

$\sqrt{22}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{22}$

Decimal Form:

$4.69041575 \dots$