Evaluate ( square root of 7+ square root of 5)^2

${(\sqrt{7} + \sqrt{5})}^{2}$

Step 1

Rewrite ${(\sqrt{7} + \sqrt{5})}^{2}$ as $(\sqrt{7} + \sqrt{5}) (\sqrt{7} + \sqrt{5})$ .

$(\sqrt{7} + \sqrt{5}) (\sqrt{7} + \sqrt{5})$

Step 2

Expand $(\sqrt{7} + \sqrt{5}) (\sqrt{7} + \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{7} (\sqrt{7} + \sqrt{5}) + \sqrt{5} (\sqrt{7} + \sqrt{5})$

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} \sqrt{5} + \sqrt{5} (\sqrt{7} + \sqrt{5})$

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} \sqrt{5} + \sqrt{5} \sqrt{7} + \sqrt{5} \sqrt{5}$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{7 \cdot 7} + \sqrt{7} \sqrt{5} + \sqrt{5} \sqrt{7} + \sqrt{5} \sqrt{5}$

Multiply $7$ by $7$ .

$\sqrt{49} + \sqrt{7} \sqrt{5} + \sqrt{5} \sqrt{7} + \sqrt{5} \sqrt{5}$

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

$\sqrt{7^{2}} + \sqrt{7} \sqrt{5} + \sqrt{5} \sqrt{7} + \sqrt{5} \sqrt{5}$

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

$7 + \sqrt{7} \sqrt{5} + \sqrt{5} \sqrt{7} + \sqrt{5} \sqrt{5}$

Combine using the product rule for radicals.

$7 + \sqrt{7 \cdot 5} + \sqrt{5} \sqrt{7} + \sqrt{5} \sqrt{5}$

Multiply $7$ by $5$ .

$7 + \sqrt{35} + \sqrt{5} \sqrt{7} + \sqrt{5} \sqrt{5}$

Combine using the product rule for radicals.

$7 + \sqrt{35} + \sqrt{5 \cdot 7} + \sqrt{5} \sqrt{5}$

Multiply $5$ by $7$ .

$7 + \sqrt{35} + \sqrt{35} + \sqrt{5} \sqrt{5}$

Combine using the product rule for radicals.

$7 + \sqrt{35} + \sqrt{35} + \sqrt{5 \cdot 5}$

Multiply $5$ by $5$ .

$7 + \sqrt{35} + \sqrt{35} + \sqrt{25}$

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

$7 + \sqrt{35} + \sqrt{35} + \sqrt{5^{2}}$

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

$7 + \sqrt{35} + \sqrt{35} + 5$

Add $7$ and $5$ .

$12 + \sqrt{35} + \sqrt{35}$

Add $\sqrt{35}$ and $\sqrt{35}$ .

$12 + 2 \sqrt{35}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$12 + 2 \sqrt{35}$

Decimal Form:

$23.83215956 \dots$

Evaluate ( square root of 7+ square root of 5)^2

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