Simplify ( square root of 5- square root of 3)/( square root of 5+ square root of 3)

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

Simplify the numerator.

Raise $\sqrt{5} - \sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{5} - \sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Step 4

Rewrite ${(\sqrt{5} - \sqrt{3})}^{2}$ as $(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})$ .

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $5$ by $5$ .

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

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

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

Multiply $- 1$ by $- 1$ .

$\frac{5 - \sqrt{15} - \sqrt{15} + 1 \sqrt{3} \sqrt{3}}{2}$

Multiply $\sqrt{3}$ by $1$ .

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

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

$\frac{5 - \sqrt{15} - \sqrt{15} + {\sqrt{3}}^{1} \sqrt{3}}{2}$

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

$\frac{5 - \sqrt{15} - \sqrt{15} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2}$

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

$\frac{5 - \sqrt{15} - \sqrt{15} + {\sqrt{3}}^{1 + 1}}{2}$

Add $1$ and $1$ .

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

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

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

$\frac{5 - \sqrt{15} - \sqrt{15} + {(3^{\frac{1}{2}})}^{2}}{2}$

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

$\frac{5 - \sqrt{15} - \sqrt{15} + 3^{\frac{1}{2} \cdot 2}}{2}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{5 - \sqrt{15} - \sqrt{15} + 3^{1}}{2}$

Evaluate the exponent.

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

Add $5$ and $3$ .

$\frac{8 - \sqrt{15} - \sqrt{15}}{2}$

Subtract $\sqrt{15}$ from $- \sqrt{15}$ .

$\frac{8 - 2 \sqrt{15}}{2}$

Step 7

Cancel the common factor of $8 - 2 \sqrt{15}$ and $2$ .

Factor $2$ out of $8$ .

$\frac{2 \cdot 4 - 2 \sqrt{15}}{2}$

Factor $2$ out of $- 2 \sqrt{15}$ .

$\frac{2 \cdot 4 + 2 (- \sqrt{15})}{2}$

Factor $2$ out of $2 (4) + 2 (- \sqrt{15})$ .

$\frac{2 (4 - \sqrt{15})}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 (4 - \sqrt{15})}{2 (1)}$

Cancel the common factor.

$\frac{2 (4 - \sqrt{15})}{2 \cdot 1}$

Rewrite the expression.

$\frac{4 - \sqrt{15}}{1}$

Divide $4 - \sqrt{15}$ by $1$ .

$4 - \sqrt{15}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$4 - \sqrt{15}$

Decimal Form:

$0.12701665 \dots$

Simplify ( square root of 5- square root of 3)/( square root of 5+ square root of 3)

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