Algebra Examples

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

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

Step 1

Rewrite

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

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

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

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

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

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

Step 2

Expand

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

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

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Step 2.1

Apply the distributive property.

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

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

Step 2.2

Apply the distributive property.

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

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

Step 2.3

Apply the distributive property.

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

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

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

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply

\sqrt{x} \sqrt{x}

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

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Step 3.1.1.1

Raise

\sqrt{x}

$\sqrt{x}$ to the power of

1

$1$ .

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

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

Step 3.1.1.2

Raise

\sqrt{x}

$\sqrt{x}$ to the power of

1

$1$ .

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

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

Step 3.1.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.1.1.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 3.1.2

Rewrite

{\sqrt{x}}^{2}

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

x

$x$ .

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Step 3.1.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

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

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

Step 3.1.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 3.1.2.4

Cancel the common factor of

$2$ .

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Step 3.1.2.4.1

Cancel the common factor.

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

Step 3.1.2.4.2

Rewrite the expression.

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

Step 3.1.2.5

Simplify.

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

Step 3.1.3

Combine using the product rule for radicals.

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

Step 3.1.4

Combine using the product rule for radicals.

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

Step 3.1.5

Combine using the product rule for radicals.

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

Step 3.1.6

Multiply

$2$ by

$2$ .

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

Step 3.1.7

Rewrite

$4$ as

$2^{2}$ .

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

Step 3.1.8

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

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

Step 3.2

Add

$\sqrt{x \cdot 2}$ and

$\sqrt{2 x}$ .

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Step 3.2.1

Reorder

$x$ and

$2$ .

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

Step 3.2.2

Add

$\sqrt{2 \cdot x}$ and

$\sqrt{2 x}$ .

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