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

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

Step 1

Simplify each term.

Multiply $\frac{4 + \sqrt{5}}{4 - \sqrt{5}}$ by $\frac{4 + \sqrt{5}}{4 + \sqrt{5}}$ .

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

Multiply $\frac{4 + \sqrt{5}}{4 - \sqrt{5}}$ by $\frac{4 + \sqrt{5}}{4 + \sqrt{5}}$ .

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

Raise $4 + \sqrt{5}$ to the power of $1$ .

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

Raise $4 + \sqrt{5}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

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

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $5$ .

$\frac{16 + 4 \sqrt{5} + 4 \sqrt{5} + \sqrt{25}}{11} + \frac{4 - \sqrt{5}}{4 + \sqrt{5}}$

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Add $16$ and $5$ .

$\frac{21 + 4 \sqrt{5} + 4 \sqrt{5}}{11} + \frac{4 - \sqrt{5}}{4 + \sqrt{5}}$

Add $4 \sqrt{5}$ and $4 \sqrt{5}$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{4 - \sqrt{5}}{4 + \sqrt{5}}$

Multiply $\frac{4 - \sqrt{5}}{4 + \sqrt{5}}$ by $\frac{4 - \sqrt{5}}{4 - \sqrt{5}}$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{4 - \sqrt{5}}{4 + \sqrt{5}} \cdot \frac{4 - \sqrt{5}}{4 - \sqrt{5}}$

Multiply $\frac{4 - \sqrt{5}}{4 + \sqrt{5}}$ by $\frac{4 - \sqrt{5}}{4 - \sqrt{5}}$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{(4 - \sqrt{5}) (4 - \sqrt{5})}{(4 + \sqrt{5}) (4 - \sqrt{5})}$

Expand the denominator using the FOIL method.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{(4 - \sqrt{5}) (4 - \sqrt{5})}{16 - 4 \sqrt{5} + \sqrt{5} \cdot 4 - {\sqrt{5}}^{2}}$

Simplify.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{(4 - \sqrt{5}) (4 - \sqrt{5})}{11}$

Simplify the numerator.

Raise $4 - \sqrt{5}$ to the power of $1$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{{(4 - \sqrt{5})}^{1} (4 - \sqrt{5})}{11}$

Raise $4 - \sqrt{5}$ to the power of $1$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{{(4 - \sqrt{5})}^{1} {(4 - \sqrt{5})}^{1}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{{(4 - \sqrt{5})}^{1 + 1}}{11}$

Add $1$ and $1$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{{(4 - \sqrt{5})}^{2}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{(4 - \sqrt{5}) (4 - \sqrt{5})}{11}$

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

Apply the distributive property.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{4 (4 - \sqrt{5}) - \sqrt{5} (4 - \sqrt{5})}{11}$

Apply the distributive property.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{4 \cdot 4 + 4 (- \sqrt{5}) - \sqrt{5} (4 - \sqrt{5})}{11}$

Apply the distributive property.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{4 \cdot 4 + 4 (- \sqrt{5}) - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})}{11}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 + 4 (- \sqrt{5}) - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})}{11}$

Multiply $- 1$ by $4$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})}{11}$

Multiply $4$ by $- 1$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} - \sqrt{5} (- \sqrt{5})}{11}$

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

Multiply $- 1$ by $- 1$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + 1 \sqrt{5} \sqrt{5}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + \sqrt{5} \sqrt{5}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1 + 1}}{11}$

Add $1$ and $1$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{2}}{11}$

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

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + {(5^{\frac{1}{2}})}^{2}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{1}{2} \cdot 2}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{2}{2}}}{11}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{2}{2}}}{11}$

Rewrite the expression.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + 5^{1}}{11}$

Evaluate the exponent.

$\frac{21 + 8 \sqrt{5}}{11} + \frac{16 - 4 \sqrt{5} - 4 \sqrt{5} + 5}{11}$

Add $16$ and $5$ .

$\frac{21 + 8 \sqrt{5}}{11} + \frac{21 - 4 \sqrt{5} - 4 \sqrt{5}}{11}$

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

$\frac{21 + 8 \sqrt{5}}{11} + \frac{21 - 8 \sqrt{5}}{11}$

Step 2

Simplify terms.

Combine the numerators over the common denominator.

$\frac{21 + 8 \sqrt{5} + 21 - 8 \sqrt{5}}{11}$

Add $21$ and $21$ .

$\frac{42 + 8 \sqrt{5} - 8 \sqrt{5}}{11}$

Subtract $8 \sqrt{5}$ from $8 \sqrt{5}$ .

$\frac{42 + 0}{11}$

Add $42$ and $0$ .

$\frac{42}{11}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{42}{11}$

Decimal Form:

$3.\overline{81}$

Mixed Number Form:

$3 \frac{9}{11}$

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

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