Evaluate (4- square root of 5)^2

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

Step 1

Rewrite ${(4 - \sqrt{5})}^{2}$ as $(4 - \sqrt{5}) (4 - \sqrt{5})$ .

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$16 + 4 (- \sqrt{5}) - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})$

Multiply $- 1$ by $4$ .

$16 - 4 \sqrt{5} - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})$

Multiply $4$ by $- 1$ .

$16 - 4 \sqrt{5} - 4 \sqrt{5} - \sqrt{5} (- \sqrt{5})$

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

Multiply $- 1$ by $- 1$ .

$16 - 4 \sqrt{5} - 4 \sqrt{5} + 1 \sqrt{5} \sqrt{5}$

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

$16 - 4 \sqrt{5} - 4 \sqrt{5} + \sqrt{5} \sqrt{5}$

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

$16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5}$

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

$16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}$

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

$16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{2}$

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

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

$16 - 4 \sqrt{5} - 4 \sqrt{5} + {(5^{\frac{1}{2}})}^{2}$

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

$16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{1}{2} \cdot 2}$

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

$16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{2}{2}}$

Rewrite the expression.

$16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{1}$

Evaluate the exponent.

$16 - 4 \sqrt{5} - 4 \sqrt{5} + 5$

Add $16$ and $5$ .

$21 - 4 \sqrt{5} - 4 \sqrt{5}$

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

$21 - 8 \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$21 - 8 \sqrt{5}$

Decimal Form:

$3.11145618 \dots$

Evaluate (4- square root of 5)^2

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