Algebra Examples

$\frac{2 - \sqrt{3}}{5 - \sqrt{3}}$

Step 1

Multiply $\frac{2 - \sqrt{3}}{5 - \sqrt{3}}$ by $\frac{5 + \sqrt{3}}{5 + \sqrt{3}}$ .

$\frac{2 - \sqrt{3}}{5 - \sqrt{3}} \cdot \frac{5 + \sqrt{3}}{5 + \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{2 - \sqrt{3}}{5 - \sqrt{3}}$ by $\frac{5 + \sqrt{3}}{5 + \sqrt{3}}$ .

$\frac{(2 - \sqrt{3}) (5 + \sqrt{3})}{(5 - \sqrt{3}) (5 + \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Expand $(2 - \sqrt{3}) (5 + \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 5 + 2 \sqrt{3} - \sqrt{3} (5 + \sqrt{3})}{22}$

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $2$ by $5$ .

$\frac{10 + 2 \sqrt{3} - \sqrt{3} \cdot 5 - \sqrt{3} \sqrt{3}}{22}$

Step 4.1.2

Multiply $5$ by $- 1$ .

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - \sqrt{3} \sqrt{3}}{22}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - ({\sqrt{3}}^{1} \sqrt{3})}{22}$

Step 4.1.3.2

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

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{22}$

Step 4.1.3.3

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

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - {\sqrt{3}}^{1 + 1}}{22}$

Step 4.1.3.4

Add $1$ and $1$ .

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - {\sqrt{3}}^{2}}{22}$

Step 4.1.4

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

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Step 4.1.4.1

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

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - {(3^{\frac{1}{2}})}^{2}}{22}$

Step 4.1.4.2

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

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - 3^{\frac{1}{2} \cdot 2}}{22}$

Step 4.1.4.3

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

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - 3^{\frac{2}{2}}}{22}$

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - 3^{\frac{2}{2}}}{22}$

Step 4.1.4.4.2

Rewrite the expression.

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - 3^{1}}{22}$

Step 4.1.4.5

Evaluate the exponent.

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - 1 \cdot 3}{22}$

Step 4.1.5

Multiply $- 1$ by $3$ .

$\frac{10 + 2 \sqrt{3} - 5 \sqrt{3} - 3}{22}$

Step 4.2

Subtract $3$ from $10$ .

$\frac{7 + 2 \sqrt{3} - 5 \sqrt{3}}{22}$

Step 4.3

Subtract $5 \sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{7 - 3 \sqrt{3}}{22}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{7 - 3 \sqrt{3}}{22}$

Decimal Form:

$0.08199307 \dots$