Algebra Examples

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

Step 1

Rewrite ${(1 + x \sqrt{2})}^{2}$ as $(1 + x \sqrt{2}) (1 + x \sqrt{2})$ .

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

Step 2

Expand $(1 + x \sqrt{2}) (1 + x \sqrt{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 3.1.2

Multiply $x \sqrt{2}$ by $1$ .

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

Step 3.1.3

Multiply $x$ by $1$ .

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

Step 3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.4.1

Move $x$ .

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

Step 3.1.4.2

Multiply $x$ by $x$ .

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

Step 3.1.5

Multiply $x^{2} \sqrt{2} \sqrt{2}$ .

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

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

Step 3.1.5.4

Add $1$ and $1$ .

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

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

Step 3.1.6.4.2

Rewrite the expression.

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

Step 3.1.6.5

Evaluate the exponent.

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

Step 3.1.7

Move $2$ to the left of $x^{2}$ .

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

Step 3.2

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

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