Simplify ( square root of x+ square root of 2)^2

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Algebra

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

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{2} + \sqrt{2} \sqrt{x} + \sqrt{2} \sqrt{2}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $\sqrt{x} \sqrt{x}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

$x^{\frac{2}{2}} + \sqrt{x} \sqrt{2} + \sqrt{2} \sqrt{x} + \sqrt{2} \sqrt{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x^{\frac{2}{2}} + \sqrt{x} \sqrt{2} + \sqrt{2} \sqrt{x} + \sqrt{2} \sqrt{2}$

Divide $1$ by $1$ .

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

Simplify.

$x + \sqrt{x} \sqrt{2} + \sqrt{2} \sqrt{x} + \sqrt{2} \sqrt{2}$

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

$x + \sqrt{x \cdot 2} + \sqrt{2 x} + \sqrt{4}$

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

$x + \sqrt{x \cdot 2} + \sqrt{2 x} + \sqrt{2^{2}}$

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

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

Add $\sqrt{x \cdot 2}$ and $\sqrt{2 x}$ .

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Reorder $x$ and $2$ .

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

Add $\sqrt{2 \cdot x}$ and $\sqrt{2 x}$ .

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

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