Simplify square root of 2( square root of 20+ square root of 8)

$\sqrt{2} (\sqrt{20} + \sqrt{8})$

Step 1

Simplify each term.

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\sqrt{2} (\sqrt{4 (5)} + \sqrt{8})$

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

$\sqrt{2} (\sqrt{2^{2} \cdot 5} + \sqrt{8})$

Pull terms out from under the radical.

$\sqrt{2} (2 \sqrt{5} + \sqrt{8})$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\sqrt{2} (2 \sqrt{5} + \sqrt{4 (2)})$

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

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

Pull terms out from under the radical.

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

Step 2

Apply the distributive property.

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

Step 3

Multiply $\sqrt{2} (2 \sqrt{5})$ .

Combine using the product rule for radicals.

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

Multiply $2$ by $5$ .

$2 \sqrt{10} + \sqrt{2} (2 \sqrt{2})$

Step 4

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

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

$2 \sqrt{10} + 2 ({\sqrt{2}}^{1} \sqrt{2})$

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

$2 \sqrt{10} + 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$2 \sqrt{10} + 2 {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

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

Step 5

Simplify each term.

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

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

$2 \sqrt{10} + 2 {(2^{\frac{1}{2}})}^{2}$

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

$2 \sqrt{10} + 2 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$2 \sqrt{10} + 2 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \sqrt{10} + 2 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$2 \sqrt{10} + 2 \cdot 2^{1}$

Evaluate the exponent.

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

Multiply $2$ by $2$ .

$2 \sqrt{10} + 4$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{10} + 4$

Decimal Form:

$10.32455532 \dots$

Simplify square root of 2( square root of 20+ square root of 8)

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