Algebra Examples

${(\sqrt{2 x^{2}} - \sqrt{5})}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Reorder $2$ and $x^{2}$ .

${(\sqrt{x^{2} \cdot 2} - \sqrt{5})}^{2}$

Step 1.1.2

Pull terms out from under the radical.

${(x \sqrt{2} - \sqrt{5})}^{2}$

Step 1.2

Rewrite ${(x \sqrt{2} - \sqrt{5})}^{2}$ as $(x \sqrt{2} - \sqrt{5}) (x \sqrt{2} - \sqrt{5})$ .

$(x \sqrt{2} - \sqrt{5}) (x \sqrt{2} - \sqrt{5})$

Step 2

Expand $(x \sqrt{2} - \sqrt{5}) (x \sqrt{2} - \sqrt{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x \sqrt{2} (x \sqrt{2} - \sqrt{5}) - \sqrt{5} (x \sqrt{2} - \sqrt{5})$

Step 2.2

Apply the distributive property.

$x \sqrt{2} (x \sqrt{2}) + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2} - \sqrt{5})$

Step 2.3

Apply the distributive property.

$x \sqrt{2} (x \sqrt{2}) + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

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

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Step 3.1.1.1

Move $x$ .

$x \cdot x \sqrt{2} \sqrt{2} + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.1.2

Multiply $x$ by $x$ .

$x^{2} \sqrt{2} \sqrt{2} + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.2

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

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Step 3.1.2.1

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

$x^{2} ({\sqrt{2}}^{1} \sqrt{2}) + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.2.2

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

$x^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.2.3

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

$x^{2} {\sqrt{2}}^{1 + 1} + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.2.4

Add $1$ and $1$ .

$x^{2} {\sqrt{2}}^{2} + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.3

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

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Step 3.1.3.1

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

$x^{2} {(2^{\frac{1}{2}})}^{2} + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.3.2

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

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

Step 3.1.3.3

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

$x^{2} \cdot 2^{\frac{2}{2}} + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.3.4.1

Cancel the common factor.

$x^{2} \cdot 2^{\frac{2}{2}} + x \sqrt{2} (- \sqrt{5}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.3.4.2

Rewrite the expression.

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

Step 3.1.3.5

Evaluate the exponent.

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

Step 3.1.4

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

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

Step 3.1.5

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

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Step 3.1.5.1

Combine using the product rule for radicals.

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

Step 3.1.5.2

Multiply $2$ by $5$ .

$2 x^{2} + x (- \sqrt{10}) - \sqrt{5} (x \sqrt{2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.6

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

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Step 3.1.6.1

Combine using the product rule for radicals.

$2 x^{2} + x (- \sqrt{10}) - 1 (x \sqrt{5 \cdot 2}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.6.2

Multiply $5$ by $2$ .

$2 x^{2} + x (- \sqrt{10}) - 1 (x \sqrt{10}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.7

Rewrite $- 1 (x \sqrt{10})$ as $- (x \sqrt{10})$ .

$2 x^{2} + x (- \sqrt{10}) - (x \sqrt{10}) - \sqrt{5} (- \sqrt{5})$

Step 3.1.8

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

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Step 3.1.8.1

Multiply $- 1$ by $- 1$ .

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + 1 \sqrt{5} \sqrt{5}$

Step 3.1.8.2

Multiply $\sqrt{5}$ by $1$ .

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + \sqrt{5} \sqrt{5}$

Step 3.1.8.3

Raise $\sqrt{5}$ to the power of $1$ .

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + {\sqrt{5}}^{1} \sqrt{5}$

Step 3.1.8.4

Raise $\sqrt{5}$ to the power of $1$ .

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}$

Step 3.1.8.5

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

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + {\sqrt{5}}^{1 + 1}$

Step 3.1.8.6

Add $1$ and $1$ .

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + {\sqrt{5}}^{2}$

Step 3.1.9

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + {(5^{\frac{1}{2}})}^{2}$

Step 3.1.9.2

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

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + 5^{\frac{1}{2} \cdot 2}$

Step 3.1.9.3

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

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + 5^{\frac{2}{2}}$

Step 3.1.9.4

Cancel the common factor of $2$ .

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Step 3.1.9.4.1

Cancel the common factor.

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + 5^{\frac{2}{2}}$

Step 3.1.9.4.2

Rewrite the expression.

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + 5^{1}$

Step 3.1.9.5

Evaluate the exponent.

$2 x^{2} + x (- \sqrt{10}) - x \sqrt{10} + 5$

Step 3.2

Subtract $x \sqrt{10}$ from $x (- \sqrt{10})$ .

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Step 3.2.1

Reorder $x$ and $- 1$ .

$2 x^{2} - 1 \cdot x \sqrt{10} - x \sqrt{10} + 5$

Step 3.2.2

Subtract $x \sqrt{10}$ from $- 1 \cdot x \sqrt{10}$ .

$2 x^{2} - 2 x \sqrt{10} + 5$