Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand the denominator using the FOIL method.

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

Step 1.4

Simplify.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

Add $1$ and $1$ .

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Apply the distributive property.

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

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $4$ by $4$ .

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

Step 1.8.1.2

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

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

Step 1.8.1.3

Combine using the product rule for radicals.

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

Step 1.8.1.4

Multiply $5$ by $5$ .

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

Step 1.8.1.5

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

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

Step 1.8.1.6

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

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

Step 1.8.2

Add $16$ and $5$ .

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

Step 1.8.3

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Expand the denominator using the FOIL method.

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

Step 1.12

Simplify.

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

Step 1.13

Simplify the numerator.

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Step 1.13.1

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

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

Step 1.13.2

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

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

Step 1.13.3

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

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

Step 1.13.4

Add $1$ and $1$ .

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

Step 1.14

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

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

Step 1.15

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

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Step 1.15.1

Apply the distributive property.

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

Step 1.15.2

Apply the distributive property.

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

Step 1.15.3

Apply the distributive property.

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

Step 1.16

Simplify and combine like terms.

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Step 1.16.1

Simplify each term.

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Step 1.16.1.1

Multiply $4$ by $4$ .

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

Step 1.16.1.2

Multiply $- 1$ by $4$ .

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

Step 1.16.1.3

Multiply $4$ by $- 1$ .

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

Step 1.16.1.4

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

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Step 1.16.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.16.1.4.2

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

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

Step 1.16.1.4.3

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

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

Step 1.16.1.4.4

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

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

Step 1.16.1.4.5

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

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

Step 1.16.1.4.6

Add $1$ and $1$ .

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

Step 1.16.1.5

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

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Step 1.16.1.5.1

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

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

Step 1.16.1.5.2

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

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

Step 1.16.1.5.3

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

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

Step 1.16.1.5.4

Cancel the common factor of $2$ .

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Step 1.16.1.5.4.1

Cancel the common factor.

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

Step 1.16.1.5.4.2

Rewrite the expression.

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

Step 1.16.1.5.5

Evaluate the exponent.

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

Step 1.16.2

Add $16$ and $5$ .

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

Step 1.16.3

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

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $21$ and $21$ .

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

Step 2.3

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

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

Step 2.4

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:

$3 \frac{9}{11}$