Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 - \sqrt{5}) (\sqrt{5} - \sqrt{3})}{{\sqrt{5}}^{2} - \sqrt{15} + \sqrt{15} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $- 1$ by $5$ .

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

Step 4.2

Multiply $- \sqrt{5} \sqrt{5}$ .

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Step 4.2.1

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 4.2.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 4.2.3

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

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

Step 4.2.4

Add $1$ and $1$ .

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

Step 4.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 4.3.2

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

$\frac{5 \sqrt{5} - 5 \sqrt{3} - 5^{\frac{1}{2} \cdot 2} - \sqrt{5} (- \sqrt{3})}{2}$

Step 4.3.3

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

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

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

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

Step 4.3.5

Evaluate the exponent.

$\frac{5 \sqrt{5} - 5 \sqrt{3} - 1 \cdot 5 - \sqrt{5} (- \sqrt{3})}{2}$

Step 4.4

Multiply $- 1$ by $5$ .

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

Step 4.5

Multiply $- \sqrt{5} (- \sqrt{3})$ .

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.5.2

Multiply $\sqrt{5}$ by $1$ .

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

Step 4.5.3

Combine using the product rule for radicals.

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

Step 4.5.4

Multiply $5$ by $3$ .

$\frac{5 \sqrt{5} - 5 \sqrt{3} - 5 + \sqrt{15}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{5} - 5 \sqrt{3} - 5 + \sqrt{15}}{2}$

Decimal Form:

$0.69653459 \dots$