Precalculus Examples

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

Step 1

Multiply $\frac{3 \sqrt{7} + 5}{\sqrt{7} - 2}$ by $\frac{\sqrt{7} + 2}{\sqrt{7} + 2}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{3 \sqrt{7} + 5}{\sqrt{7} - 2}$ by $\frac{\sqrt{7} + 2}{\sqrt{7} + 2}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Expand $(3 \sqrt{7} + 5) (\sqrt{7} + 2)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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 $3 \sqrt{7} \sqrt{7}$ .

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

$\frac{3 {\sqrt{7}}^{1 + 1} + 3 \sqrt{7} \cdot 2 + 5 \sqrt{7} + 5 \cdot 2}{3}$

Step 4.1.1.4

Add $1$ and $1$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

$\frac{3 \cdot 7^{\frac{1}{2} \cdot 2} + 3 \sqrt{7} \cdot 2 + 5 \sqrt{7} + 5 \cdot 2}{3}$

Step 4.1.2.3

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

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

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

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

Step 4.1.2.4.2

Rewrite the expression.

$\frac{3 \cdot 7^{1} + 3 \sqrt{7} \cdot 2 + 5 \sqrt{7} + 5 \cdot 2}{3}$

Step 4.1.2.5

Evaluate the exponent.

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

Step 4.1.3

Multiply $3$ by $7$ .

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

Step 4.1.4

Multiply $2$ by $3$ .

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

Step 4.1.5

Multiply $5$ by $2$ .

$\frac{21 + 6 \sqrt{7} + 5 \sqrt{7} + 10}{3}$

Step 4.2

Add $21$ and $10$ .

$\frac{31 + 6 \sqrt{7} + 5 \sqrt{7}}{3}$

Step 4.3

Add $6 \sqrt{7}$ and $5 \sqrt{7}$ .

$\frac{31 + 11 \sqrt{7}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{31 + 11 \sqrt{7}}{3}$

Decimal Form:

$20.03442147 \dots$