Evaluate ( square root of 17+ square root of 13)/( square root of 17- square root of 13)

$\frac{\sqrt{17} + \sqrt{13}}{\sqrt{17} - \sqrt{13}}$

Step 1

Multiply $\frac{\sqrt{17} + \sqrt{13}}{\sqrt{17} - \sqrt{13}}$ by $\frac{\sqrt{17} + \sqrt{13}}{\sqrt{17} + \sqrt{13}}$ .

$\frac{\sqrt{17} + \sqrt{13}}{\sqrt{17} - \sqrt{13}} \cdot \frac{\sqrt{17} + \sqrt{13}}{\sqrt{17} + \sqrt{13}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{17} + \sqrt{13}}{\sqrt{17} - \sqrt{13}}$ by $\frac{\sqrt{17} + \sqrt{13}}{\sqrt{17} + \sqrt{13}}$ .

$\frac{(\sqrt{17} + \sqrt{13}) (\sqrt{17} + \sqrt{13})}{(\sqrt{17} - \sqrt{13}) (\sqrt{17} + \sqrt{13})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{17} + \sqrt{13}) (\sqrt{17} + \sqrt{13})}{{\sqrt{17}}^{2} + \sqrt{221} - \sqrt{221} - {\sqrt{13}}^{2}}$

Simplify.

$\frac{(\sqrt{17} + \sqrt{13}) (\sqrt{17} + \sqrt{13})}{4}$

Step 3

Simplify the numerator.

Raise $\sqrt{17} + \sqrt{13}$ to the power of $1$ .

$\frac{{(\sqrt{17} + \sqrt{13})}^{1} (\sqrt{17} + \sqrt{13})}{4}$

Raise $\sqrt{17} + \sqrt{13}$ to the power of $1$ .

$\frac{{(\sqrt{17} + \sqrt{13})}^{1} {(\sqrt{17} + \sqrt{13})}^{1}}{4}$

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

$\frac{{(\sqrt{17} + \sqrt{13})}^{1 + 1}}{4}$

Add $1$ and $1$ .

$\frac{{(\sqrt{17} + \sqrt{13})}^{2}}{4}$

Step 4

Rewrite ${(\sqrt{17} + \sqrt{13})}^{2}$ as $(\sqrt{17} + \sqrt{13}) (\sqrt{17} + \sqrt{13})$ .

$\frac{(\sqrt{17} + \sqrt{13}) (\sqrt{17} + \sqrt{13})}{4}$

Step 5

Expand $(\sqrt{17} + \sqrt{13}) (\sqrt{17} + \sqrt{13})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{17} (\sqrt{17} + \sqrt{13}) + \sqrt{13} (\sqrt{17} + \sqrt{13})}{4}$

Apply the distributive property.

$\frac{\sqrt{17} \sqrt{17} + \sqrt{17} \sqrt{13} + \sqrt{13} (\sqrt{17} + \sqrt{13})}{4}$

Apply the distributive property.

$\frac{\sqrt{17} \sqrt{17} + \sqrt{17} \sqrt{13} + \sqrt{13} \sqrt{17} + \sqrt{13} \sqrt{13}}{4}$

Step 6

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{17 \cdot 17} + \sqrt{17} \sqrt{13} + \sqrt{13} \sqrt{17} + \sqrt{13} \sqrt{13}}{4}$

Multiply $17$ by $17$ .

$\frac{\sqrt{289} + \sqrt{17} \sqrt{13} + \sqrt{13} \sqrt{17} + \sqrt{13} \sqrt{13}}{4}$

Rewrite $289$ as $17^{2}$ .

$\frac{\sqrt{17^{2}} + \sqrt{17} \sqrt{13} + \sqrt{13} \sqrt{17} + \sqrt{13} \sqrt{13}}{4}$

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

$\frac{17 + \sqrt{17} \sqrt{13} + \sqrt{13} \sqrt{17} + \sqrt{13} \sqrt{13}}{4}$

Combine using the product rule for radicals.

$\frac{17 + \sqrt{17 \cdot 13} + \sqrt{13} \sqrt{17} + \sqrt{13} \sqrt{13}}{4}$

Multiply $17$ by $13$ .

$\frac{17 + \sqrt{221} + \sqrt{13} \sqrt{17} + \sqrt{13} \sqrt{13}}{4}$

Combine using the product rule for radicals.

$\frac{17 + \sqrt{221} + \sqrt{13 \cdot 17} + \sqrt{13} \sqrt{13}}{4}$

Multiply $13$ by $17$ .

$\frac{17 + \sqrt{221} + \sqrt{221} + \sqrt{13} \sqrt{13}}{4}$

Combine using the product rule for radicals.

$\frac{17 + \sqrt{221} + \sqrt{221} + \sqrt{13 \cdot 13}}{4}$

Multiply $13$ by $13$ .

$\frac{17 + \sqrt{221} + \sqrt{221} + \sqrt{169}}{4}$

Rewrite $169$ as $13^{2}$ .

$\frac{17 + \sqrt{221} + \sqrt{221} + \sqrt{13^{2}}}{4}$

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

$\frac{17 + \sqrt{221} + \sqrt{221} + 13}{4}$

Add $17$ and $13$ .

$\frac{30 + \sqrt{221} + \sqrt{221}}{4}$

Add $\sqrt{221}$ and $\sqrt{221}$ .

$\frac{30 + 2 \sqrt{221}}{4}$

Step 7

Cancel the common factor of $30 + 2 \sqrt{221}$ and $4$ .

Factor $2$ out of $30$ .

$\frac{2 \cdot 15 + 2 \sqrt{221}}{4}$

Factor $2$ out of $2 \sqrt{221}$ .

$\frac{2 \cdot 15 + 2 (\sqrt{221})}{4}$

Factor $2$ out of $2 \cdot 15 + 2 (\sqrt{221})$ .

$\frac{2 \cdot (15 + \sqrt{221})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 \cdot (15 + \sqrt{221})}{2 (2)}$

Cancel the common factor.

$\frac{2 \cdot (15 + \sqrt{221})}{2 \cdot 2}$

Rewrite the expression.

$\frac{15 + \sqrt{221}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{15 + \sqrt{221}}{2}$

Decimal Form:

$14.93303437 \dots$

Evaluate ( square root of 17+ square root of 13)/( square root of 17- square root of 13)

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