Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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

Multiply $2$ by $2$ .

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

Step 2.1.2

Multiply $- 1$ by $2$ .

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

Step 2.1.3

Multiply $2$ by $- 1$ .

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

Step 2.1.4

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

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Step 2.1.4.1

Multiply $- 1$ by $- 1$ .

$4 - 2 \sqrt{5} - 2 \sqrt{5} + 1 \sqrt{5} \sqrt{5}$

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

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

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

Step 2.1.4.5

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

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

Step 2.1.4.6

Add $1$ and $1$ .

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

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

Step 2.1.5.4.2

Rewrite the expression.

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

Step 2.1.5.5

Evaluate the exponent.

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

Step 2.2

Add $4$ and $5$ .

$9 - 2 \sqrt{5} - 2 \sqrt{5}$

Step 2.3

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

$9 - 4 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$9 - 4 \sqrt{5}$

Decimal Form:

$0.05572809 \dots$