Algebra Examples

$\frac{\sqrt{7} + \sqrt{3}}{\sqrt{12} + \sqrt{7}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1

Factor $4$ out of $12$ .

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

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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

Step 4

Expand $(\sqrt{7} + \sqrt{3}) (2 \sqrt{3} - \sqrt{7})$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $\sqrt{7} (2 \sqrt{3})$ .

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Step 5.1.1.1

Combine using the product rule for radicals.

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

Step 5.1.1.2

Multiply $7$ by $3$ .

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

Step 5.1.2

Multiply $\sqrt{7} (- \sqrt{7})$ .

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Step 5.1.2.1

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

$\frac{2 \sqrt{21} - ({\sqrt{7}}^{1} \sqrt{7}) + \sqrt{3} (2 \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.2.2

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

$\frac{2 \sqrt{21} - ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) + \sqrt{3} (2 \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.2.3

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

$\frac{2 \sqrt{21} - {\sqrt{7}}^{1 + 1} + \sqrt{3} (2 \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.2.4

Add $1$ and $1$ .

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

Step 5.1.3

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{2 \sqrt{21} - {(7^{\frac{1}{2}})}^{2} + \sqrt{3} (2 \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.3.2

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

$\frac{2 \sqrt{21} - 7^{\frac{1}{2} \cdot 2} + \sqrt{3} (2 \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.3.3

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

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

Step 5.1.3.4

Cancel the common factor of $2$ .

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Step 5.1.3.4.1

Cancel the common factor.

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

Step 5.1.3.4.2

Rewrite the expression.

$\frac{2 \sqrt{21} - 7^{1} + \sqrt{3} (2 \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.3.5

Evaluate the exponent.

$\frac{2 \sqrt{21} - 1 \cdot 7 + \sqrt{3} (2 \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.4

Multiply $- 1$ by $7$ .

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

Step 5.1.5

Multiply $\sqrt{3} (2 \sqrt{3})$ .

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Step 5.1.5.1

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

$\frac{2 \sqrt{21} - 7 + 2 ({\sqrt{3}}^{1} \sqrt{3}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.5.2

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

$\frac{2 \sqrt{21} - 7 + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.5.3

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

$\frac{2 \sqrt{21} - 7 + 2 {\sqrt{3}}^{1 + 1} + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.5.4

Add $1$ and $1$ .

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

Step 5.1.6

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

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Step 5.1.6.1

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

$\frac{2 \sqrt{21} - 7 + 2 {(3^{\frac{1}{2}})}^{2} + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.6.2

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

$\frac{2 \sqrt{21} - 7 + 2 \cdot 3^{\frac{1}{2} \cdot 2} + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.6.3

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

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

Step 5.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.6.4.1

Cancel the common factor.

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

Step 5.1.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{21} - 7 + 2 \cdot 3^{1} + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.6.5

Evaluate the exponent.

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

Step 5.1.7

Multiply $2$ by $3$ .

$\frac{2 \sqrt{21} - 7 + 6 + \sqrt{3} (- \sqrt{7})}{5}$

Step 5.1.8

Multiply $\sqrt{3} (- \sqrt{7})$ .

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Step 5.1.8.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{21} - 7 + 6 - \sqrt{3 \cdot 7}}{5}$

Step 5.1.8.2

Multiply $3$ by $7$ .

$\frac{2 \sqrt{21} - 7 + 6 - \sqrt{21}}{5}$

Step 5.2

Subtract $\sqrt{21}$ from $2 \sqrt{21}$ .

$\frac{- 7 + 6 + \sqrt{21}}{5}$

Step 5.3

Add $- 7$ and $6$ .

$\frac{- 1 + \sqrt{21}}{5}$

Step 6

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) + \sqrt{21}}{5}$

Step 7

Factor $- 1$ out of $\sqrt{21}$ .

$\frac{- 1 (1) - 1 (- \sqrt{21})}{5}$

Step 8

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{21})$ .

$\frac{- 1 (1 - \sqrt{21})}{5}$

Step 9

Move the negative in front of the fraction.

$- \frac{1 - \sqrt{21}}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{1 - \sqrt{21}}{5}$

Decimal Form:

$0.71651513 \dots$