Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

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

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

Step 7

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

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Combine using the product rule for radicals.

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

Step 8.1.2

Multiply $7$ by $7$ .

$\frac{\sqrt{49} + \sqrt{7} \sqrt{3} + \sqrt{3} \sqrt{7} + \sqrt{3} \sqrt{3}}{4}$

Step 8.1.3

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

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

Step 8.1.4

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

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

Step 8.1.5

Combine using the product rule for radicals.

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

Step 8.1.6

Multiply $7$ by $3$ .

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

Step 8.1.7

Combine using the product rule for radicals.

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

Step 8.1.8

Multiply $3$ by $7$ .

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

Step 8.1.9

Combine using the product rule for radicals.

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

Step 8.1.10

Multiply $3$ by $3$ .

$\frac{7 + \sqrt{21} + \sqrt{21} + \sqrt{9}}{4}$

Step 8.1.11

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

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

Step 8.1.12

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

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

Step 8.2

Add $7$ and $3$ .

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

Step 8.3

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

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

Step 9

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

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Step 9.1

Factor $2$ out of $10$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $2$ out of $4$ .

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

Step 9.4.2

Cancel the common factor.

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

Step 9.4.3

Rewrite the expression.

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

Step 10

Split the fraction $\frac{5 + \sqrt{21}}{2}$ into two fractions.

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$4.79128784 \dots$