Precalculus Examples

$\frac{\frac{7}{\sqrt{113}} + \frac{3}{\sqrt{73}}}{1 + (\frac{7}{\sqrt{113}}) (\frac{3}{\sqrt{73}})}$

Step 1

Multiply the numerator and denominator of the fraction by $\sqrt{113} \sqrt{73}$ .

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Step 1.1

Multiply $\frac{\frac{7}{\sqrt{113}} + \frac{3}{\sqrt{73}}}{1 + \frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}}}$ by $\frac{\sqrt{113} \sqrt{73}}{\sqrt{113} \sqrt{73}}$ .

$\frac{\sqrt{113} \sqrt{73}}{\sqrt{113} \sqrt{73}} \cdot \frac{\frac{7}{\sqrt{113}} + \frac{3}{\sqrt{73}}}{1 + \frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}}}$

Step 1.2

Combine.

$\frac{\sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} + \frac{3}{\sqrt{73}})}{\sqrt{113} \sqrt{73} (1 + \frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 2

Apply the distributive property.

$\frac{\sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}} + \sqrt{113} \sqrt{73} \frac{3}{\sqrt{73}}}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $\sqrt{113}$ .

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Step 3.1.1

Factor $\sqrt{113}$ out of $\sqrt{113} \sqrt{73}$ .

$\frac{\sqrt{113} (\sqrt{73}) \frac{7}{\sqrt{113}} + \sqrt{113} \sqrt{73} \frac{3}{\sqrt{73}}}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 3.1.2

Cancel the common factor.

$\frac{\sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}} + \sqrt{113} \sqrt{73} \frac{3}{\sqrt{73}}}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 3.1.3

Rewrite the expression.

$\frac{\sqrt{73} \cdot 7 + \sqrt{113} \sqrt{73} \frac{3}{\sqrt{73}}}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 3.2

Cancel the common factor of $\sqrt{73}$ .

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Step 3.2.1

Factor $\sqrt{73}$ out of $\sqrt{113} \sqrt{73}$ .

$\frac{\sqrt{73} \cdot 7 + \sqrt{73} \sqrt{113} \frac{3}{\sqrt{73}}}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 3.2.2

Cancel the common factor.

$\frac{\sqrt{73} \cdot 7 + \sqrt{73} \sqrt{113} \frac{3}{\sqrt{73}}}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 3.2.3

Rewrite the expression.

$\frac{\sqrt{73} \cdot 7 + \sqrt{113} \cdot 3}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} (\frac{7}{\sqrt{113}} \cdot \frac{3}{\sqrt{73}})}$

Step 4

Simplify the numerator.

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Step 4.1

Move $7$ to the left of $\sqrt{73}$ .

$\frac{7 \cdot \sqrt{73} + \sqrt{113} \cdot 3}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}} \frac{3}{\sqrt{73}}}$

Step 4.2

Move $3$ to the left of $\sqrt{113}$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{113} \sqrt{73} \cdot 1 + \sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}} \frac{3}{\sqrt{73}}}$

Step 5

Simplify the denominator.

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Step 5.1

Combine using the product rule for radicals.

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{113 \cdot 73} \cdot 1 + \sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}} \frac{3}{\sqrt{73}}}$

Step 5.2

Multiply $113$ by $73$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} \cdot 1 + \sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}} \frac{3}{\sqrt{73}}}$

Step 5.3

Multiply $\sqrt{8249}$ by $1$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}} \frac{3}{\sqrt{73}}}$

Step 5.4

Cancel the common factor of $\sqrt{73}$ .

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Step 5.4.1

Factor $\sqrt{73}$ out of $\sqrt{113} \sqrt{73} \frac{7}{\sqrt{113}}$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \sqrt{73} (\sqrt{113} \frac{7}{\sqrt{113}}) \frac{3}{\sqrt{73}}}$

Step 5.4.2

Cancel the common factor.

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \sqrt{73} (\sqrt{113} \frac{7}{\sqrt{113}}) \frac{3}{\sqrt{73}}}$

Step 5.4.3

Rewrite the expression.

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \sqrt{113} \frac{7}{\sqrt{113}} \cdot 3}$

Step 5.5

Combine $\sqrt{113}$ and $\frac{7}{\sqrt{113}}$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \frac{\sqrt{113} \cdot 7}{\sqrt{113}} \cdot 3}$

Step 5.6

Combine $\frac{\sqrt{113} \cdot 7}{\sqrt{113}}$ and $3$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \frac{\sqrt{113} \cdot 7 \cdot 3}{\sqrt{113}}}$

Step 5.7

Multiply $3$ by $7$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \frac{\sqrt{113} \cdot 21}{\sqrt{113}}}$

Step 5.8

Cancel the common factor of $\sqrt{113}$ .

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Step 5.8.1

Cancel the common factor.

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + \frac{\sqrt{113} \cdot 21}{\sqrt{113}}}$

Step 5.8.2

Divide $21$ by $1$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + 21}$

Step 6

Multiply $\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + 21}$ by $\frac{\sqrt{8249} - 21}{\sqrt{8249} - 21}$ .

$\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + 21} \cdot \frac{\sqrt{8249} - 21}{\sqrt{8249} - 21}$

Step 7

Combine fractions.

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Step 7.1

Multiply $\frac{7 \sqrt{73} + 3 \sqrt{113}}{\sqrt{8249} + 21}$ by $\frac{\sqrt{8249} - 21}{\sqrt{8249} - 21}$ .

$\frac{(7 \sqrt{73} + 3 \sqrt{113}) (\sqrt{8249} - 21)}{(\sqrt{8249} + 21) (\sqrt{8249} - 21)}$

Step 7.2

Expand the denominator using the FOIL method.

$\frac{(7 \sqrt{73} + 3 \sqrt{113}) (\sqrt{8249} - 21)}{{\sqrt{8249}}^{2} + \sqrt{8249} \cdot - 21 + 21 \sqrt{8249} - 441}$

Step 7.3

Simplify.

$\frac{(7 \sqrt{73} + 3 \sqrt{113}) (\sqrt{8249} - 21)}{7808}$

Step 8

Expand $(7 \sqrt{73} + 3 \sqrt{113}) (\sqrt{8249} - 21)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$\frac{7 \sqrt{73} (\sqrt{8249} - 21) + 3 \sqrt{113} (\sqrt{8249} - 21)}{7808}$

Step 8.2

Apply the distributive property.

$\frac{7 \sqrt{73} \sqrt{8249} + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} (\sqrt{8249} - 21)}{7808}$

Step 8.3

Apply the distributive property.

$\frac{7 \sqrt{73} \sqrt{8249} + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $7 \sqrt{73} \sqrt{8249}$ .

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Step 9.1.1.1

Combine using the product rule for radicals.

$\frac{7 \sqrt{8249 \cdot 73} + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.1.2

Multiply $8249$ by $73$ .

$\frac{7 \sqrt{602177} + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.2

Rewrite $602177$ as $73^{2} \cdot 113$ .

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Step 9.1.2.1

Factor $5329$ out of $602177$ .

$\frac{7 \sqrt{5329 (113)} + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.2.2

Rewrite $5329$ as $73^{2}$ .

$\frac{7 \sqrt{73^{2} \cdot 113} + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.3

Pull terms out from under the radical.

$\frac{7 (73 \sqrt{113}) + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.4

Multiply $73$ by $7$ .

$\frac{511 \sqrt{113} + 7 \sqrt{73} \cdot - 21 + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.5

Multiply $- 21$ by $7$ .

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 3 \sqrt{113} \sqrt{8249} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.6

Multiply $3 \sqrt{113} \sqrt{8249}$ .

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Step 9.1.6.1

Combine using the product rule for radicals.

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 3 \sqrt{8249 \cdot 113} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.6.2

Multiply $8249$ by $113$ .

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 3 \sqrt{932137} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.7

Rewrite $932137$ as $113^{2} \cdot 73$ .

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Step 9.1.7.1

Factor $12769$ out of $932137$ .

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 3 \sqrt{12769 (73)} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.7.2

Rewrite $12769$ as $113^{2}$ .

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 3 \sqrt{113^{2} \cdot 73} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.8

Pull terms out from under the radical.

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 3 (113 \sqrt{73}) + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.9

Multiply $113$ by $3$ .

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 339 \sqrt{73} + 3 \sqrt{113} \cdot - 21}{7808}$

Step 9.1.10

Multiply $- 21$ by $3$ .

$\frac{511 \sqrt{113} - 147 \sqrt{73} + 339 \sqrt{73} - 63 \sqrt{113}}{7808}$

Step 9.2

Subtract $63 \sqrt{113}$ from $511 \sqrt{113}$ .

$\frac{448 \sqrt{113} - 147 \sqrt{73} + 339 \sqrt{73}}{7808}$

Step 9.3

Add $- 147 \sqrt{73}$ and $339 \sqrt{73}$ .

$\frac{448 \sqrt{113} + 192 \sqrt{73}}{7808}$

Step 10

Cancel the common factor of $448 \sqrt{113} + 192 \sqrt{73}$ and $7808$ .

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Step 10.1

Factor $64$ out of $448 \sqrt{113}$ .

$\frac{64 (7 \sqrt{113}) + 192 \sqrt{73}}{7808}$

Step 10.2

Factor $64$ out of $192 \sqrt{73}$ .

$\frac{64 (7 \sqrt{113}) + 64 (3 \sqrt{73})}{7808}$

Step 10.3

Factor $64$ out of $64 (7 \sqrt{113}) + 64 (3 \sqrt{73})$ .

$\frac{64 (7 \sqrt{113} + 3 \sqrt{73})}{7808}$

Step 10.4

Cancel the common factors.

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Step 10.4.1

Factor $64$ out of $7808$ .

$\frac{64 (7 \sqrt{113} + 3 \sqrt{73})}{64 (122)}$

Step 10.4.2

Cancel the common factor.

$\frac{64 (7 \sqrt{113} + 3 \sqrt{73})}{64 \cdot 122}$

Step 10.4.3

Rewrite the expression.

$\frac{7 \sqrt{113} + 3 \sqrt{73}}{122}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{113} + 3 \sqrt{73}}{122}$

Decimal Form:

$0.82002485 \dots$