Simplify the Radical Expression ( square root of 5+ square root of 2)^2

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

Step 1

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $5$ by $5$ .

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

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

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

Pull terms out from under the radical.

$| 5 | + \sqrt{5} \sqrt{2} + \sqrt{2} \sqrt{5} + \sqrt{2} \sqrt{2}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

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

Combine using the product rule for radicals.

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

Multiply $5$ by $2$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $5$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

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

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

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

Pull terms out from under the radical.

$5 + \sqrt{10} + \sqrt{10} + | 2 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Add $5$ and $2$ .

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

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

$7 + 2 \sqrt{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$7 + 2 \sqrt{10}$

Decimal Form:

$13.32455532 \dots$

Simplify the Radical Expression ( square root of 5+ square root of 2)^2

Related Questions

Support LCMGCF.com by disabling your adblocker.