Algebra Examples

$\frac{5 - \sqrt{21}}{\sqrt{3} - \sqrt{7}}$

Step 1

Multiply $\frac{5 - \sqrt{21}}{\sqrt{3} - \sqrt{7}}$ by $\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}}$ .

$\frac{5 - \sqrt{21}}{\sqrt{3} - \sqrt{7}} \cdot \frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 - \sqrt{21}}{\sqrt{3} - \sqrt{7}}$ by $\frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}}$ .

$\frac{(5 - \sqrt{21}) (\sqrt{3} + \sqrt{7})}{(\sqrt{3} - \sqrt{7}) (\sqrt{3} + \sqrt{7})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 - \sqrt{21}) (\sqrt{3} + \sqrt{7})}{{\sqrt{3}}^{2} + \sqrt{21} - \sqrt{21} - {\sqrt{7}}^{2}}$

Step 2.3

Simplify.

$\frac{(5 - \sqrt{21}) (\sqrt{3} + \sqrt{7})}{- 4}$

Step 3

Expand $(5 - \sqrt{21}) (\sqrt{3} + \sqrt{7})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{5 (\sqrt{3} + \sqrt{7}) - \sqrt{21} (\sqrt{3} + \sqrt{7})}{- 4}$

Step 3.2

Apply the distributive property.

$\frac{5 \sqrt{3} + 5 \sqrt{7} - \sqrt{21} (\sqrt{3} + \sqrt{7})}{- 4}$

Step 3.3

Apply the distributive property.

$\frac{5 \sqrt{3} + 5 \sqrt{7} - \sqrt{21} \sqrt{3} - \sqrt{21} \sqrt{7}}{- 4}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $- \sqrt{21} \sqrt{3}$ .

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Step 4.1.1.1

Combine using the product rule for radicals.

$\frac{5 \sqrt{3} + 5 \sqrt{7} - \sqrt{3 \cdot 21} - \sqrt{21} \sqrt{7}}{- 4}$

Step 4.1.1.2

Multiply $3$ by $21$ .

$\frac{5 \sqrt{3} + 5 \sqrt{7} - \sqrt{63} - \sqrt{21} \sqrt{7}}{- 4}$

Step 4.1.2

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 4.1.2.1

Factor $9$ out of $63$ .

$\frac{5 \sqrt{3} + 5 \sqrt{7} - \sqrt{9 (7)} - \sqrt{21} \sqrt{7}}{- 4}$

Step 4.1.2.2

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

$\frac{5 \sqrt{3} + 5 \sqrt{7} - \sqrt{3^{2} \cdot 7} - \sqrt{21} \sqrt{7}}{- 4}$

Step 4.1.3

Pull terms out from under the radical.

$\frac{5 \sqrt{3} + 5 \sqrt{7} - (3 \sqrt{7}) - \sqrt{21} \sqrt{7}}{- 4}$

Step 4.1.4

Multiply $3$ by $- 1$ .

$\frac{5 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7} - \sqrt{21} \sqrt{7}}{- 4}$

Step 4.1.5

Multiply $- \sqrt{21} \sqrt{7}$ .

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Step 4.1.5.1

Combine using the product rule for radicals.

$\frac{5 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7} - \sqrt{7 \cdot 21}}{- 4}$

Step 4.1.5.2

Multiply $7$ by $21$ .

$\frac{5 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7} - \sqrt{147}}{- 4}$

Step 4.1.6

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 4.1.6.1

Factor $49$ out of $147$ .

$\frac{5 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7} - \sqrt{49 (3)}}{- 4}$

Step 4.1.6.2

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

$\frac{5 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7} - \sqrt{7^{2} \cdot 3}}{- 4}$

Step 4.1.7

Pull terms out from under the radical.

$\frac{5 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7} - (7 \sqrt{3})}{- 4}$

Step 4.1.8

Multiply $7$ by $- 1$ .

$\frac{5 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7} - 7 \sqrt{3}}{- 4}$

Step 4.2

Subtract $7 \sqrt{3}$ from $5 \sqrt{3}$ .

$\frac{- 2 \sqrt{3} + 5 \sqrt{7} - 3 \sqrt{7}}{- 4}$

Step 4.3

Subtract $3 \sqrt{7}$ from $5 \sqrt{7}$ .

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

Step 5

Cancel the common factor of $- 2 \sqrt{3} + 2 \sqrt{7}$ and $- 4$ .

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Step 5.1

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 5.2

Factor $2$ out of $2 \sqrt{7}$ .

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

Step 5.3

Factor $2$ out of $2 (- \sqrt{3}) + 2 (\sqrt{7})$ .

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

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $2$ out of $- 4$ .

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

Step 5.4.2

Cancel the common factor.

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

Step 5.4.3

Rewrite the expression.

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

Step 6

Move the negative in front of the fraction.

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

Step 7

Factor $- 1$ out of $- \sqrt{3}$ .

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

Step 8

Factor $- 1$ out of $\sqrt{7}$ .

$- \frac{- (\sqrt{3}) - 1 (- \sqrt{7})}{2}$

Step 9

Factor $- 1$ out of $- (\sqrt{3}) - 1 (- \sqrt{7})$ .

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

Step 10

Simplify the expression.

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Step 10.1

Rewrite $- (\sqrt{3} - \sqrt{7})$ as $- 1 (\sqrt{3} - \sqrt{7})$ .

$- \frac{- 1 (\sqrt{3} - \sqrt{7})}{2}$

Step 10.2

Move the negative in front of the fraction.

$- - \frac{\sqrt{3} - \sqrt{7}}{2}$

Step 10.3

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{3} - \sqrt{7}}{2}$

Step 10.4

Multiply $\frac{\sqrt{3} - \sqrt{7}}{2}$ by $1$ .

$\frac{\sqrt{3} - \sqrt{7}}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3} - \sqrt{7}}{2}$

Decimal Form:

$- 0.45685025 \dots$