Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{x} \sqrt{x} + \sqrt{x} (- \sqrt{5}) + \sqrt{5} \sqrt{x} + \sqrt{5} (- \sqrt{5})$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{x} (- \sqrt{5})$ and $\sqrt{5} \sqrt{x}$ .

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

Step 2.1.2

Add $- \sqrt{5} \sqrt{x}$ and $\sqrt{5} \sqrt{x}$ .

$\sqrt{x} \sqrt{x} + 0 + \sqrt{5} (- \sqrt{5})$

Step 2.1.3

Add $\sqrt{x} \sqrt{x}$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

Step 2.2.2.4.2

Rewrite the expression.

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

Step 2.2.2.5

Simplify.

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

$x - {\sqrt{5}}^{1 + 1}$

Step 2.2.3.4

Add $1$ and $1$ .

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

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

$x - 5^{\frac{1}{2} \cdot 2}$

Step 2.2.4.3

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

$x - 5^{\frac{2}{2}}$

Step 2.2.4.4

Cancel the common factor of $2$ .

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Step 2.2.4.4.1

Cancel the common factor.

$x - 5^{\frac{2}{2}}$

Step 2.2.4.4.2

Rewrite the expression.

$x - 5^{1}$

Step 2.2.4.5

Evaluate the exponent.

$x - 1 \cdot 5$

Step 2.2.5

Multiply $- 1$ by $5$ .

$x - 5$