Algebra Examples

$\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ by $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ .

$\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.2

Multiply $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ by $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ .

$\frac{(\sqrt{5} + \sqrt{3}) (\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + \sqrt{3})} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} + \sqrt{3}) (\sqrt{5} + \sqrt{3})}{{\sqrt{5}}^{2} + \sqrt{15} - \sqrt{15} - {\sqrt{3}}^{2}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.4

Simplify.

$\frac{(\sqrt{5} + \sqrt{3}) (\sqrt{5} + \sqrt{3})}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Raise $\sqrt{5} + \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{5} + \sqrt{3})}^{1} (\sqrt{5} + \sqrt{3})}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.5.2

Raise $\sqrt{5} + \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{5} + \sqrt{3})}^{1} {(\sqrt{5} + \sqrt{3})}^{1}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.5.3

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

$\frac{{(\sqrt{5} + \sqrt{3})}^{1 + 1}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.5.4

Add $1$ and $1$ .

$\frac{{(\sqrt{5} + \sqrt{3})}^{2}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.6

Rewrite ${(\sqrt{5} + \sqrt{3})}^{2}$ as $(\sqrt{5} + \sqrt{3}) (\sqrt{5} + \sqrt{3})$ .

$\frac{(\sqrt{5} + \sqrt{3}) (\sqrt{5} + \sqrt{3})}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.7

Expand $(\sqrt{5} + \sqrt{3}) (\sqrt{5} + \sqrt{3})$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

$\frac{\sqrt{5} (\sqrt{5} + \sqrt{3}) + \sqrt{3} (\sqrt{5} + \sqrt{3})}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.7.2

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{5} + \sqrt{5} \sqrt{3} + \sqrt{3} (\sqrt{5} + \sqrt{3})}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.7.3

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{5} + \sqrt{5} \sqrt{3} + \sqrt{3} \sqrt{5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

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

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 5} + \sqrt{5} \sqrt{3} + \sqrt{3} \sqrt{5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.2

Multiply $5$ by $5$ .

$\frac{\sqrt{25} + \sqrt{5} \sqrt{3} + \sqrt{3} \sqrt{5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.3

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

$\frac{\sqrt{5^{2}} + \sqrt{5} \sqrt{3} + \sqrt{3} \sqrt{5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.4

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

$\frac{5 + \sqrt{5} \sqrt{3} + \sqrt{3} \sqrt{5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.5

Combine using the product rule for radicals.

$\frac{5 + \sqrt{5 \cdot 3} + \sqrt{3} \sqrt{5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.6

Multiply $5$ by $3$ .

$\frac{5 + \sqrt{15} + \sqrt{3} \sqrt{5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.7

Combine using the product rule for radicals.

$\frac{5 + \sqrt{15} + \sqrt{3 \cdot 5} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.8

Multiply $3$ by $5$ .

$\frac{5 + \sqrt{15} + \sqrt{15} + \sqrt{3} \sqrt{3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.9

Combine using the product rule for radicals.

$\frac{5 + \sqrt{15} + \sqrt{15} + \sqrt{3 \cdot 3}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.10

Multiply $3$ by $3$ .

$\frac{5 + \sqrt{15} + \sqrt{15} + \sqrt{9}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.11

Rewrite $9$ as $3^{2}$ .

$\frac{5 + \sqrt{15} + \sqrt{15} + \sqrt{3^{2}}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.1.12

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

$\frac{5 + \sqrt{15} + \sqrt{15} + 3}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.2

Add $5$ and $3$ .

$\frac{8 + \sqrt{15} + \sqrt{15}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.8.3

Add $\sqrt{15}$ and $\sqrt{15}$ .

$\frac{8 + 2 \sqrt{15}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.9

Cancel the common factor of $8 + 2 \sqrt{15}$ and $2$ .

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Step 1.9.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4 + 2 \sqrt{15}}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.9.2

Factor $2$ out of $2 \sqrt{15}$ .

$\frac{2 \cdot 4 + 2 (\sqrt{15})}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.9.3

Factor $2$ out of $2 (4) + 2 (\sqrt{15})$ .

$\frac{2 (4 + \sqrt{15})}{2} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.9.4

Cancel the common factors.

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Step 1.9.4.1

Factor $2$ out of $2$ .

$\frac{2 (4 + \sqrt{15})}{2 (1)} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.9.4.2

Cancel the common factor.

$\frac{2 (4 + \sqrt{15})}{2 \cdot 1} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.9.4.3

Rewrite the expression.

$\frac{4 + \sqrt{15}}{1} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.9.4.4

Divide $4 + \sqrt{15}$ by $1$ .

$4 + \sqrt{15} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$

Step 1.10

Multiply $\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ by $\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ .

$4 + \sqrt{15} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \cdot \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$

Step 1.11

Multiply $\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ by $\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ .

$4 + \sqrt{15} + \frac{(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3})}$

Step 1.12

Expand the denominator using the FOIL method.

$4 + \sqrt{15} + \frac{(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})}{{\sqrt{5}}^{2} - \sqrt{15} + \sqrt{15} - {\sqrt{3}}^{2}}$

Step 1.13

Simplify.

$4 + \sqrt{15} + \frac{(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})}{2}$

Step 1.14

Simplify the numerator.

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Step 1.14.1

Raise $\sqrt{5} - \sqrt{3}$ to the power of $1$ .

$4 + \sqrt{15} + \frac{{(\sqrt{5} - \sqrt{3})}^{1} (\sqrt{5} - \sqrt{3})}{2}$

Step 1.14.2

Raise $\sqrt{5} - \sqrt{3}$ to the power of $1$ .

$4 + \sqrt{15} + \frac{{(\sqrt{5} - \sqrt{3})}^{1} {(\sqrt{5} - \sqrt{3})}^{1}}{2}$

Step 1.14.3

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

$4 + \sqrt{15} + \frac{{(\sqrt{5} - \sqrt{3})}^{1 + 1}}{2}$

Step 1.14.4

Add $1$ and $1$ .

$4 + \sqrt{15} + \frac{{(\sqrt{5} - \sqrt{3})}^{2}}{2}$

Step 1.15

Rewrite ${(\sqrt{5} - \sqrt{3})}^{2}$ as $(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})$ .

$4 + \sqrt{15} + \frac{(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})}{2}$

Step 1.16

Expand $(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})$ using the FOIL Method.

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Step 1.16.1

Apply the distributive property.

$4 + \sqrt{15} + \frac{\sqrt{5} (\sqrt{5} - \sqrt{3}) - \sqrt{3} (\sqrt{5} - \sqrt{3})}{2}$

Step 1.16.2

Apply the distributive property.

$4 + \sqrt{15} + \frac{\sqrt{5} \sqrt{5} + \sqrt{5} (- \sqrt{3}) - \sqrt{3} (\sqrt{5} - \sqrt{3})}{2}$

Step 1.16.3

Apply the distributive property.

$4 + \sqrt{15} + \frac{\sqrt{5} \sqrt{5} + \sqrt{5} (- \sqrt{3}) - \sqrt{3} \sqrt{5} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17

Simplify and combine like terms.

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Step 1.17.1

Simplify each term.

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Step 1.17.1.1

Combine using the product rule for radicals.

$4 + \sqrt{15} + \frac{\sqrt{5 \cdot 5} + \sqrt{5} (- \sqrt{3}) - \sqrt{3} \sqrt{5} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.2

Multiply $5$ by $5$ .

$4 + \sqrt{15} + \frac{\sqrt{25} + \sqrt{5} (- \sqrt{3}) - \sqrt{3} \sqrt{5} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.3

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

$4 + \sqrt{15} + \frac{\sqrt{5^{2}} + \sqrt{5} (- \sqrt{3}) - \sqrt{3} \sqrt{5} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.4

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

$4 + \sqrt{15} + \frac{5 + \sqrt{5} (- \sqrt{3}) - \sqrt{3} \sqrt{5} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.5

Multiply $\sqrt{5} (- \sqrt{3})$ .

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Step 1.17.1.5.1

Combine using the product rule for radicals.

$4 + \sqrt{15} + \frac{5 - \sqrt{5 \cdot 3} - \sqrt{3} \sqrt{5} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.5.2

Multiply $5$ by $3$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{3} \sqrt{5} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.6

Multiply $- \sqrt{3} \sqrt{5}$ .

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Step 1.17.1.6.1

Combine using the product rule for radicals.

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{5 \cdot 3} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.6.2

Multiply $5$ by $3$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} - \sqrt{3} (- \sqrt{3})}{2}$

Step 1.17.1.7

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 1.17.1.7.1

Multiply $- 1$ by $- 1$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + 1 \sqrt{3} \sqrt{3}}{2}$

Step 1.17.1.7.2

Multiply $\sqrt{3}$ by $1$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + \sqrt{3} \sqrt{3}}{2}$

Step 1.17.1.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + {\sqrt{3}}^{1} \sqrt{3}}{2}$

Step 1.17.1.7.4

Raise $\sqrt{3}$ to the power of $1$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2}$

Step 1.17.1.7.5

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

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + {\sqrt{3}}^{1 + 1}}{2}$

Step 1.17.1.7.6

Add $1$ and $1$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + {\sqrt{3}}^{2}}{2}$

Step 1.17.1.8

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.17.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + {(3^{\frac{1}{2}})}^{2}}{2}$

Step 1.17.1.8.2

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

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + 3^{\frac{1}{2} \cdot 2}}{2}$

Step 1.17.1.8.3

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

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + 3^{\frac{2}{2}}}{2}$

Step 1.17.1.8.4

Cancel the common factor of $2$ .

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Step 1.17.1.8.4.1

Cancel the common factor.

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + 3^{\frac{2}{2}}}{2}$

Step 1.17.1.8.4.2

Rewrite the expression.

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + 3^{1}}{2}$

Step 1.17.1.8.5

Evaluate the exponent.

$4 + \sqrt{15} + \frac{5 - \sqrt{15} - \sqrt{15} + 3}{2}$

Step 1.17.2

Add $5$ and $3$ .

$4 + \sqrt{15} + \frac{8 - \sqrt{15} - \sqrt{15}}{2}$

Step 1.17.3

Subtract $\sqrt{15}$ from $- \sqrt{15}$ .

$4 + \sqrt{15} + \frac{8 - 2 \sqrt{15}}{2}$

Step 1.18

Cancel the common factor of $8 - 2 \sqrt{15}$ and $2$ .

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Step 1.18.1

Factor $2$ out of $8$ .

$4 + \sqrt{15} + \frac{2 \cdot 4 - 2 \sqrt{15}}{2}$

Step 1.18.2

Factor $2$ out of $- 2 \sqrt{15}$ .

$4 + \sqrt{15} + \frac{2 \cdot 4 + 2 (- \sqrt{15})}{2}$

Step 1.18.3

Factor $2$ out of $2 (4) + 2 (- \sqrt{15})$ .

$4 + \sqrt{15} + \frac{2 (4 - \sqrt{15})}{2}$

Step 1.18.4

Cancel the common factors.

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Step 1.18.4.1

Factor $2$ out of $2$ .

$4 + \sqrt{15} + \frac{2 (4 - \sqrt{15})}{2 (1)}$

Step 1.18.4.2

Cancel the common factor.

$4 + \sqrt{15} + \frac{2 (4 - \sqrt{15})}{2 \cdot 1}$

Step 1.18.4.3

Rewrite the expression.

$4 + \sqrt{15} + \frac{4 - \sqrt{15}}{1}$

Step 1.18.4.4

Divide $4 - \sqrt{15}$ by $1$ .

$4 + \sqrt{15} + 4 - \sqrt{15}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $4$ and $4$ .

$8 + \sqrt{15} - \sqrt{15}$

Step 2.2

Subtract $\sqrt{15}$ from $\sqrt{15}$ .

$8 + 0$

Step 2.3

Add $8$ and $0$ .

$8$