Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} \sqrt{5} + \sqrt{5} \sqrt{x} + \sqrt{5} \sqrt{5}$

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

$x^{\frac{2}{2}} + \sqrt{x} \sqrt{5} + \sqrt{5} \sqrt{x} + \sqrt{5} \sqrt{5}$

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.

$x^{\frac{2}{2}} + \sqrt{x} \sqrt{5} + \sqrt{5} \sqrt{x} + \sqrt{5} \sqrt{5}$

Step 3.1.2.4.2

Rewrite the expression.

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

Step 3.1.2.5

Simplify.

$x + \sqrt{x} \sqrt{5} + \sqrt{5} \sqrt{x} + \sqrt{5} \sqrt{5}$

Step 3.1.3

Combine using the product rule for radicals.

$x + \sqrt{x \cdot 5} + \sqrt{5} \sqrt{x} + \sqrt{5} \sqrt{5}$

Step 3.1.4

Combine using the product rule for radicals.

$x + \sqrt{x \cdot 5} + \sqrt{5 x} + \sqrt{5} \sqrt{5}$

Step 3.1.5

Combine using the product rule for radicals.

$x + \sqrt{x \cdot 5} + \sqrt{5 x} + \sqrt{5 \cdot 5}$

Step 3.1.6

Multiply $5$ by $5$ .

$x + \sqrt{x \cdot 5} + \sqrt{5 x} + \sqrt{25}$

Step 3.1.7

Rewrite $25$ as $5^{2}$ .

$x + \sqrt{x \cdot 5} + \sqrt{5 x} + \sqrt{5^{2}}$

Step 3.1.8

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

$x + \sqrt{x \cdot 5} + \sqrt{5 x} + 5$

Step 3.2

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

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Step 3.2.1

Reorder $x$ and $5$ .

$x + \sqrt{5 \cdot x} + \sqrt{5 x} + 5$

Step 3.2.2

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

$x + 2 \sqrt{5 \cdot x} + 5$