Simplify (2+ square root of 14)/( square root of 7+ square root of 2)

$\frac{2 + \sqrt{14}}{\sqrt{7} + \sqrt{2}}$

Step 1

Multiply $\frac{2 + \sqrt{14}}{\sqrt{7} + \sqrt{2}}$ by $\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}$ .

$\frac{2 + \sqrt{14}}{\sqrt{7} + \sqrt{2}} \cdot \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}$

Step 2

Combine fractions.

Multiply $\frac{2 + \sqrt{14}}{\sqrt{7} + \sqrt{2}}$ by $\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}$ .

$\frac{(2 + \sqrt{14}) (\sqrt{7} - \sqrt{2})}{(\sqrt{7} + \sqrt{2}) (\sqrt{7} - \sqrt{2})}$

Expand the denominator using the FOIL method.

$\frac{(2 + \sqrt{14}) (\sqrt{7} - \sqrt{2})}{{\sqrt{7}}^{2} - \sqrt{14} + \sqrt{14} - {\sqrt{2}}^{2}}$

Simplify.

$\frac{(2 + \sqrt{14}) (\sqrt{7} - \sqrt{2})}{5}$

Step 3

Simplify the numerator.

Expand $(2 + \sqrt{14}) (\sqrt{7} - \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (\sqrt{7} - \sqrt{2}) + \sqrt{14} (\sqrt{7} - \sqrt{2})}{5}$

Apply the distributive property.

$\frac{2 \sqrt{7} + 2 (- \sqrt{2}) + \sqrt{14} (\sqrt{7} - \sqrt{2})}{5}$

Apply the distributive property.

$\frac{2 \sqrt{7} + 2 (- \sqrt{2}) + \sqrt{14} \sqrt{7} + \sqrt{14} (- \sqrt{2})}{5}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{7} - 2 \sqrt{2} + \sqrt{14} \sqrt{7} + \sqrt{14} (- \sqrt{2})}{5}$

Combine using the product rule for radicals.

$\frac{2 \sqrt{7} - 2 \sqrt{2} + \sqrt{14 \cdot 7} + \sqrt{14} (- \sqrt{2})}{5}$

Multiply $14$ by $7$ .

$\frac{2 \sqrt{7} - 2 \sqrt{2} + \sqrt{98} + \sqrt{14} (- \sqrt{2})}{5}$

Rewrite $98$ as $7^{2} \cdot 2$ .

Factor $49$ out of $98$ .

$\frac{2 \sqrt{7} - 2 \sqrt{2} + \sqrt{49 (2)} + \sqrt{14} (- \sqrt{2})}{5}$

Rewrite $49$ as $7^{2}$ .

$\frac{2 \sqrt{7} - 2 \sqrt{2} + \sqrt{7^{2} \cdot 2} + \sqrt{14} (- \sqrt{2})}{5}$

Pull terms out from under the radical.

$\frac{2 \sqrt{7} - 2 \sqrt{2} + 7 \sqrt{2} + \sqrt{14} (- \sqrt{2})}{5}$

Multiply $\sqrt{14} (- \sqrt{2})$ .

Combine using the product rule for radicals.

$\frac{2 \sqrt{7} - 2 \sqrt{2} + 7 \sqrt{2} - \sqrt{14 \cdot 2}}{5}$

Multiply $14$ by $2$ .

$\frac{2 \sqrt{7} - 2 \sqrt{2} + 7 \sqrt{2} - \sqrt{28}}{5}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$\frac{2 \sqrt{7} - 2 \sqrt{2} + 7 \sqrt{2} - \sqrt{4 (7)}}{5}$

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

$\frac{2 \sqrt{7} - 2 \sqrt{2} + 7 \sqrt{2} - \sqrt{2^{2} \cdot 7}}{5}$

Pull terms out from under the radical.

$\frac{2 \sqrt{7} - 2 \sqrt{2} + 7 \sqrt{2} - (2 \sqrt{7})}{5}$

Multiply $2$ by $- 1$ .

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

Subtract $2 \sqrt{7}$ from $2 \sqrt{7}$ .

$\frac{0 - 2 \sqrt{2} + 7 \sqrt{2}}{5}$

Subtract $2 \sqrt{2}$ from $0$ .

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

Add $- 2 \sqrt{2}$ and $7 \sqrt{2}$ .

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

Step 4

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$

Simplify (2+ square root of 14)/( square root of 7+ square root of 2)

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