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

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

Step 1

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

Multiply $2$ by $2$ .

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

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

$4 ({\sqrt{x}}^{1} \sqrt{x}) + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

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

$4 ({\sqrt{x}}^{1} {\sqrt{x}}^{1}) + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

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

$4 {\sqrt{x}}^{1 + 1} + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

Add $1$ and $1$ .

$4 {\sqrt{x}}^{2} + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

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

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

$4 {(x^{\frac{1}{2}})}^{2} + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

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

$4 x^{\frac{1}{2} \cdot 2} + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

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

$4 x^{\frac{2}{2}} + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x^{\frac{2}{2}} + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

Rewrite the expression.

$4 x^{1} + 2 \sqrt{x} \sqrt{2} + \sqrt{2} (2 \sqrt{x}) + \sqrt{2} \sqrt{2}$

Simplify.

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

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

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