Algebra Examples

$(\sqrt{5} + 2) (4 - \sqrt{5})$

Step 1

Expand $(\sqrt{5} + 2) (4 - \sqrt{5})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{5} (4 - \sqrt{5}) + 2 (4 - \sqrt{5})$

Step 1.2

Apply the distributive property.

$\sqrt{5} \cdot 4 + \sqrt{5} (- \sqrt{5}) + 2 (4 - \sqrt{5})$

Step 1.3

Apply the distributive property.

$\sqrt{5} \cdot 4 + \sqrt{5} (- \sqrt{5}) + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Move $4$ to the left of $\sqrt{5}$ .

$4 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{5}) + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.2

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

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Step 2.1.2.1

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

$4 \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5}) + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.2.2

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

$4 \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.2.3

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

$4 \sqrt{5} - {\sqrt{5}}^{1 + 1} + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.2.4

Add $1$ and $1$ .

$4 \sqrt{5} - {\sqrt{5}}^{2} + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.3

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

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Step 2.1.3.1

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

$4 \sqrt{5} - {(5^{\frac{1}{2}})}^{2} + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.3.2

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

$4 \sqrt{5} - 5^{\frac{1}{2} \cdot 2} + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.3.3

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

$4 \sqrt{5} - 5^{\frac{2}{2}} + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Cancel the common factor.

$4 \sqrt{5} - 5^{\frac{2}{2}} + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.3.4.2

Rewrite the expression.

$4 \sqrt{5} - 5^{1} + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.3.5

Evaluate the exponent.

$4 \sqrt{5} - 1 \cdot 5 + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.4

Multiply $- 1$ by $5$ .

$4 \sqrt{5} - 5 + 2 \cdot 4 + 2 (- \sqrt{5})$

Step 2.1.5

Multiply $2$ by $4$ .

$4 \sqrt{5} - 5 + 8 + 2 (- \sqrt{5})$

Step 2.1.6

Multiply $- 1$ by $2$ .

$4 \sqrt{5} - 5 + 8 - 2 \sqrt{5}$

Step 2.2

Subtract $2 \sqrt{5}$ from $4 \sqrt{5}$ .

$- 5 + 8 + 2 \sqrt{5}$

Step 2.3

Add $- 5$ and $8$ .

$3 + 2 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 + 2 \sqrt{5}$

Decimal Form:

$7.47213595 \dots$