Simplify (2-5^(1/2))*(2+ square root of 5)

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

Step 1

Expand $(2 - 5^{\frac{1}{2}}) (2 + \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- 5^{\frac{1}{2}} \sqrt{5}$ .

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $4$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1$