Simplify square root of 2( square root of 50+7)

$\sqrt{2} (\sqrt{50} + 7)$

Step 1

Simplify each term.

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$\sqrt{2} (\sqrt{25 (2)} + 7)$

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

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

Pull terms out from under the radical.

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

Step 2

Apply the distributive property.

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

Step 3

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

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

$5 ({\sqrt{2}}^{1} \sqrt{2}) + \sqrt{2} \cdot 7$

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

$5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + \sqrt{2} \cdot 7$

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

$5 {\sqrt{2}}^{1 + 1} + \sqrt{2} \cdot 7$

Add $1$ and $1$ .

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

Step 4

Move $7$ to the left of $\sqrt{2}$ .

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

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 $2^{\frac{1}{2}}$ .

$5 {(2^{\frac{1}{2}})}^{2} + 7 \cdot \sqrt{2}$

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

$5 \cdot 2^{\frac{1}{2} \cdot 2} + 7 \cdot \sqrt{2}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$5 \cdot 2^{1} + 7 \cdot \sqrt{2}$

Evaluate the exponent.

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

Multiply $5$ by $2$ .

$10 + 7 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$10 + 7 \sqrt{2}$

Decimal Form:

$19.89949493 \dots$

Simplify square root of 2( square root of 50+7)

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