Simplify ( square root of 5+ square root of 2)/( square root of 5- square root of 2)

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

Simplify the numerator.

Raise $\sqrt{5} + \sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{5} + \sqrt{2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $5$ by $5$ .

$\frac{\sqrt{25} + \sqrt{5} \sqrt{2} + \sqrt{2} \sqrt{5} + \sqrt{2} \sqrt{2}}{3}$

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $2$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $5$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

$\frac{5 + \sqrt{10} + \sqrt{10} + \sqrt{4}}{3}$

Rewrite $4$ as $2^{2}$ .

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

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

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

Add $5$ and $2$ .

$\frac{7 + \sqrt{10} + \sqrt{10}}{3}$

Add $\sqrt{10}$ and $\sqrt{10}$ .

$\frac{7 + 2 \sqrt{10}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7 + 2 \sqrt{10}}{3}$

Decimal Form:

$4.44151844 \dots$

Simplify ( square root of 5+ square root of 2)/( square root of 5- square root of 2)

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