Algebra Examples

$\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} - \sqrt{7}}$

Step 1

Multiply $\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} - \sqrt{7}}$ by $\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}}$ .

$\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} - \sqrt{7}} \cdot \frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} - \sqrt{7}}$ by $\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}}$ .

$\frac{(\sqrt{3} + \sqrt{7}) (\sqrt{3} + \sqrt{7})}{(\sqrt{3} - \sqrt{7}) (\sqrt{3} + \sqrt{7})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} + \sqrt{7}) (\sqrt{3} + \sqrt{7})}{{\sqrt{3}}^{2} + \sqrt{21} - \sqrt{21} - {\sqrt{7}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{3} + \sqrt{7}) (\sqrt{3} + \sqrt{7})}{- 4}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{3} + \sqrt{7}$ to the power of $1$ .

$\frac{{(\sqrt{3} + \sqrt{7})}^{1} (\sqrt{3} + \sqrt{7})}{- 4}$

Step 3.2

Raise $\sqrt{3} + \sqrt{7}$ to the power of $1$ .

$\frac{{(\sqrt{3} + \sqrt{7})}^{1} {(\sqrt{3} + \sqrt{7})}^{1}}{- 4}$

Step 3.3

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

$\frac{{(\sqrt{3} + \sqrt{7})}^{1 + 1}}{- 4}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(\sqrt{3} + \sqrt{7})}^{2}}{- 4}$

Step 4

Rewrite ${(\sqrt{3} + \sqrt{7})}^{2}$ as $(\sqrt{3} + \sqrt{7}) (\sqrt{3} + \sqrt{7})$ .

$\frac{(\sqrt{3} + \sqrt{7}) (\sqrt{3} + \sqrt{7})}{- 4}$

Step 5

Expand $(\sqrt{3} + \sqrt{7}) (\sqrt{3} + \sqrt{7})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{3} (\sqrt{3} + \sqrt{7}) + \sqrt{7} (\sqrt{3} + \sqrt{7})}{- 4}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{3} + \sqrt{3} \sqrt{7} + \sqrt{7} (\sqrt{3} + \sqrt{7})}{- 4}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{3} + \sqrt{3} \sqrt{7} + \sqrt{7} \sqrt{3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 3} + \sqrt{3} \sqrt{7} + \sqrt{7} \sqrt{3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.2

Multiply $3$ by $3$ .

$\frac{\sqrt{9} + \sqrt{3} \sqrt{7} + \sqrt{7} \sqrt{3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.3

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

$\frac{\sqrt{3^{2}} + \sqrt{3} \sqrt{7} + \sqrt{7} \sqrt{3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.4

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

$\frac{3 + \sqrt{3} \sqrt{7} + \sqrt{7} \sqrt{3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.5

Combine using the product rule for radicals.

$\frac{3 + \sqrt{3 \cdot 7} + \sqrt{7} \sqrt{3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.6

Multiply $3$ by $7$ .

$\frac{3 + \sqrt{21} + \sqrt{7} \sqrt{3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.7

Combine using the product rule for radicals.

$\frac{3 + \sqrt{21} + \sqrt{7 \cdot 3} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.8

Multiply $7$ by $3$ .

$\frac{3 + \sqrt{21} + \sqrt{21} + \sqrt{7} \sqrt{7}}{- 4}$

Step 6.1.9

Combine using the product rule for radicals.

$\frac{3 + \sqrt{21} + \sqrt{21} + \sqrt{7 \cdot 7}}{- 4}$

Step 6.1.10

Multiply $7$ by $7$ .

$\frac{3 + \sqrt{21} + \sqrt{21} + \sqrt{49}}{- 4}$

Step 6.1.11

Rewrite $49$ as $7^{2}$ .

$\frac{3 + \sqrt{21} + \sqrt{21} + \sqrt{7^{2}}}{- 4}$

Step 6.1.12

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

$\frac{3 + \sqrt{21} + \sqrt{21} + 7}{- 4}$

Step 6.2

Add $3$ and $7$ .

$\frac{10 + \sqrt{21} + \sqrt{21}}{- 4}$

Step 6.3

Add $\sqrt{21}$ and $\sqrt{21}$ .

$\frac{10 + 2 \sqrt{21}}{- 4}$

Step 7

Cancel the common factor of $10 + 2 \sqrt{21}$ and $- 4$ .

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Step 7.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5 + 2 \sqrt{21}}{- 4}$

Step 7.2

Factor $2$ out of $2 \sqrt{21}$ .

$\frac{2 \cdot 5 + 2 (\sqrt{21})}{- 4}$

Step 7.3

Factor $2$ out of $2 \cdot 5 + 2 (\sqrt{21})$ .

$\frac{2 \cdot (5 + \sqrt{21})}{- 4}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $2$ out of $- 4$ .

$\frac{2 \cdot (5 + \sqrt{21})}{2 (- 2)}$

Step 7.4.2

Cancel the common factor.

$\frac{2 \cdot (5 + \sqrt{21})}{2 \cdot - 2}$

Step 7.4.3

Rewrite the expression.

$\frac{5 + \sqrt{21}}{- 2}$

Step 8

Move the negative in front of the fraction.

$- \frac{5 + \sqrt{21}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 + \sqrt{21}}{2}$

Decimal Form:

$- 4.79128784 \dots$