Linear Algebra Examples

$\frac{\sqrt{23} + 4}{\sqrt{23} - 4}$

Step 1

Multiply $\frac{\sqrt{23} + 4}{\sqrt{23} - 4}$ by $\frac{\sqrt{23} + 4}{\sqrt{23} + 4}$ .

$\frac{\sqrt{23} + 4}{\sqrt{23} - 4} \cdot \frac{\sqrt{23} + 4}{\sqrt{23} + 4}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{23} + 4}{\sqrt{23} - 4}$ by $\frac{\sqrt{23} + 4}{\sqrt{23} + 4}$ .

$\frac{(\sqrt{23} + 4) (\sqrt{23} + 4)}{(\sqrt{23} - 4) (\sqrt{23} + 4)}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{23} + 4) (\sqrt{23} + 4)}{{\sqrt{23}}^{2} + \sqrt{23} \cdot 4 - 4 \sqrt{23} - 16}$

Step 2.3

Simplify.

$\frac{(\sqrt{23} + 4) (\sqrt{23} + 4)}{7}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{23} + 4$ to the power of $1$ .

$\frac{{(\sqrt{23} + 4)}^{1} (\sqrt{23} + 4)}{7}$

Step 3.2

Raise $\sqrt{23} + 4$ to the power of $1$ .

$\frac{{(\sqrt{23} + 4)}^{1} {(\sqrt{23} + 4)}^{1}}{7}$

Step 3.3

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

$\frac{{(\sqrt{23} + 4)}^{1 + 1}}{7}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(\sqrt{23} + 4)}^{2}}{7}$

Step 4

Simplify ${(\sqrt{23} + 4)}^{2}$ .

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Step 4.1

Rewrite ${(\sqrt{23} + 4)}^{2}$ as $(\sqrt{23} + 4) (\sqrt{23} + 4)$ .

$\frac{(\sqrt{23} + 4) (\sqrt{23} + 4)}{7}$

Step 4.2

Expand $(\sqrt{23} + 4) (\sqrt{23} + 4)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$\frac{\sqrt{23} (\sqrt{23} + 4) + 4 (\sqrt{23} + 4)}{7}$

Step 4.2.2

Apply the distributive property.

$\frac{\sqrt{23} \sqrt{23} + \sqrt{23} \cdot 4 + 4 (\sqrt{23} + 4)}{7}$

Step 4.2.3

Apply the distributive property.

$\frac{\sqrt{23} \sqrt{23} + \sqrt{23} \cdot 4 + 4 \sqrt{23} + 4 \cdot 4}{7}$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{23 \cdot 23} + \sqrt{23} \cdot 4 + 4 \sqrt{23} + 4 \cdot 4}{7}$

Step 4.3.1.2

Multiply $23$ by $23$ .

$\frac{\sqrt{529} + \sqrt{23} \cdot 4 + 4 \sqrt{23} + 4 \cdot 4}{7}$

Step 4.3.1.3

Rewrite $529$ as $23^{2}$ .

$\frac{\sqrt{23^{2}} + \sqrt{23} \cdot 4 + 4 \sqrt{23} + 4 \cdot 4}{7}$

Step 4.3.1.4

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

$\frac{23 + \sqrt{23} \cdot 4 + 4 \sqrt{23} + 4 \cdot 4}{7}$

Step 4.3.1.5

Move $4$ to the left of $\sqrt{23}$ .

$\frac{23 + 4 \cdot \sqrt{23} + 4 \sqrt{23} + 4 \cdot 4}{7}$

Step 4.3.1.6

Multiply $4$ by $4$ .

$\frac{23 + 4 \sqrt{23} + 4 \sqrt{23} + 16}{7}$

Step 4.3.2

Add $23$ and $16$ .

$\frac{39 + 4 \sqrt{23} + 4 \sqrt{23}}{7}$

Step 4.3.3

Add $4 \sqrt{23}$ and $4 \sqrt{23}$ .

$\frac{39 + 8 \sqrt{23}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{39 + 8 \sqrt{23}}{7}$

Decimal Form:

$11.05237888 \dots$