Simplify ((4- square root of 5)(4+ square root of 5))/(2 square root of 11)

$\frac{(4 - \sqrt{5}) (4 + \sqrt{5})}{2 \sqrt{11}}$

Step 1

Simplify the numerator.

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

Apply the distributive property.

$\frac{4 (4 + \sqrt{5}) - \sqrt{5} (4 + \sqrt{5})}{2 \sqrt{11}}$

Apply the distributive property.

$\frac{4 \cdot 4 + 4 \sqrt{5} - \sqrt{5} (4 + \sqrt{5})}{2 \sqrt{11}}$

Apply the distributive property.

$\frac{4 \cdot 4 + 4 \sqrt{5} - \sqrt{5} \cdot 4 - \sqrt{5} \sqrt{5}}{2 \sqrt{11}}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$\frac{16 + 4 \sqrt{5} - \sqrt{5} \cdot 4 - \sqrt{5} \sqrt{5}}{2 \sqrt{11}}$

Multiply $4$ by $- 1$ .

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - \sqrt{5} \sqrt{5}}{2 \sqrt{11}}$

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

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

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})}{2 \sqrt{11}}$

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

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{2 \sqrt{11}}$

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

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - {\sqrt{5}}^{1 + 1}}{2 \sqrt{11}}$

Add $1$ and $1$ .

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - {\sqrt{5}}^{2}}{2 \sqrt{11}}$

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

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

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - {(5^{\frac{1}{2}})}^{2}}{2 \sqrt{11}}$

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

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{2 \sqrt{11}}$

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

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - 5^{\frac{2}{2}}}{2 \sqrt{11}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - 5^{\frac{2}{2}}}{2 \sqrt{11}}$

Rewrite the expression.

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - 5^{1}}{2 \sqrt{11}}$

Evaluate the exponent.

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - 1 \cdot 5}{2 \sqrt{11}}$

Multiply $- 1$ by $5$ .

$\frac{16 + 4 \sqrt{5} - 4 \sqrt{5} - 5}{2 \sqrt{11}}$

Subtract $5$ from $16$ .

$\frac{11 + 4 \sqrt{5} - 4 \sqrt{5}}{2 \sqrt{11}}$

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

$\frac{11 + 0}{2 \sqrt{11}}$

Add $11$ and $0$ .

$\frac{11}{2 \sqrt{11}}$

Step 2

Multiply $\frac{11}{2 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{11}{2 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{11}{2 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{11 \sqrt{11}}{2 \sqrt{11} \sqrt{11}}$

Move $\sqrt{11}$ .

$\frac{11 \sqrt{11}}{2 (\sqrt{11} \sqrt{11})}$

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{11 \sqrt{11}}{2 ({\sqrt{11}}^{1} \sqrt{11})}$

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{11 \sqrt{11}}{2 ({\sqrt{11}}^{1} {\sqrt{11}}^{1})}$

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

$\frac{11 \sqrt{11}}{2 {\sqrt{11}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{11 \sqrt{11}}{2 {\sqrt{11}}^{2}}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{11 \sqrt{11}}{2 {(11^{\frac{1}{2}})}^{2}}$

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

$\frac{11 \sqrt{11}}{2 \cdot 11^{\frac{1}{2} \cdot 2}}$

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

$\frac{11 \sqrt{11}}{2 \cdot 11^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{11 \sqrt{11}}{2 \cdot 11^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{11 \sqrt{11}}{2 \cdot 11^{1}}$

Evaluate the exponent.

$\frac{11 \sqrt{11}}{2 \cdot 11}$

Step 4

Cancel the common factor of $11$ .

Cancel the common factor.

$\frac{11 \sqrt{11}}{2 \cdot 11}$

Rewrite the expression.

$\frac{\sqrt{11}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{11}}{2}$

Decimal Form:

$1.65831239 \dots$

Simplify ((4- square root of 5)(4+ square root of 5))/(2 square root of 11)

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