Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Add $15$ and $2$ .

$\frac{\sqrt{15} - 2}{\sqrt{17}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.2

Multiply $\frac{\sqrt{15} - 2}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{\sqrt{15} - 2}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{15} - 2}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{(\sqrt{15} - 2) \sqrt{17}}{\sqrt{17} \sqrt{17}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.2

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

$\frac{(\sqrt{15} - 2) \sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.3

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

$\frac{(\sqrt{15} - 2) \sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.4

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

$\frac{(\sqrt{15} - 2) \sqrt{17}}{{\sqrt{17}}^{1 + 1}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{(\sqrt{15} - 2) \sqrt{17}}{{\sqrt{17}}^{2}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\frac{(\sqrt{15} - 2) \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.6.2

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

$\frac{(\sqrt{15} - 2) \sqrt{17}}{17^{\frac{1}{2} \cdot 2}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.6.3

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

$\frac{(\sqrt{15} - 2) \sqrt{17}}{17^{\frac{2}{2}}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{(\sqrt{15} - 2) \sqrt{17}}{17^{\frac{2}{2}}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{(\sqrt{15} - 2) \sqrt{17}}{17^{1}} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.3.6.5

Evaluate the exponent.

$\frac{(\sqrt{15} - 2) \sqrt{17}}{17} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.4

Apply the distributive property.

$\frac{\sqrt{15} \sqrt{17} - 2 \sqrt{17}}{17} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.5

Combine using the product rule for radicals.

$\frac{\sqrt{15 \cdot 17} - 2 \sqrt{17}}{17} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.6

Multiply $15$ by $17$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2}$

Step 1.7

Multiply $\frac{\sqrt{15} + 2}{\sqrt{15} - 2}$ by $\frac{\sqrt{15} + 2}{\sqrt{15} + 2}$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{15} + 2}{\sqrt{15} - 2} \cdot \frac{\sqrt{15} + 2}{\sqrt{15} + 2}$

Step 1.8

Multiply $\frac{\sqrt{15} + 2}{\sqrt{15} - 2}$ by $\frac{\sqrt{15} + 2}{\sqrt{15} + 2}$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{(\sqrt{15} + 2) (\sqrt{15} + 2)}{(\sqrt{15} - 2) (\sqrt{15} + 2)}$

Step 1.9

Expand the denominator using the FOIL method.

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{(\sqrt{15} + 2) (\sqrt{15} + 2)}{{\sqrt{15}}^{2} + \sqrt{15} \cdot 2 - 2 \sqrt{15} - 4}$

Step 1.10

Simplify.

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{(\sqrt{15} + 2) (\sqrt{15} + 2)}{11}$

Step 1.11

Simplify the numerator.

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Step 1.11.1

Raise $\sqrt{15} + 2$ to the power of $1$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{{(\sqrt{15} + 2)}^{1} (\sqrt{15} + 2)}{11}$

Step 1.11.2

Raise $\sqrt{15} + 2$ to the power of $1$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{{(\sqrt{15} + 2)}^{1} {(\sqrt{15} + 2)}^{1}}{11}$

Step 1.11.3

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

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{{(\sqrt{15} + 2)}^{1 + 1}}{11}$

Step 1.11.4

Add $1$ and $1$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{{(\sqrt{15} + 2)}^{2}}{11}$

Step 1.12

Simplify ${(\sqrt{15} + 2)}^{2}$ .

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Step 1.12.1

Rewrite ${(\sqrt{15} + 2)}^{2}$ as $(\sqrt{15} + 2) (\sqrt{15} + 2)$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{(\sqrt{15} + 2) (\sqrt{15} + 2)}{11}$

Step 1.12.2

Expand $(\sqrt{15} + 2) (\sqrt{15} + 2)$ using the FOIL Method.

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Step 1.12.2.1

Apply the distributive property.

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{15} (\sqrt{15} + 2) + 2 (\sqrt{15} + 2)}{11}$

Step 1.12.2.2

Apply the distributive property.

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{15} \sqrt{15} + \sqrt{15} \cdot 2 + 2 (\sqrt{15} + 2)}{11}$

Step 1.12.2.3

Apply the distributive property.

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{15} \sqrt{15} + \sqrt{15} \cdot 2 + 2 \sqrt{15} + 2 \cdot 2}{11}$

Step 1.12.3

Simplify and combine like terms.

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Step 1.12.3.1

Simplify each term.

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Step 1.12.3.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{15 \cdot 15} + \sqrt{15} \cdot 2 + 2 \sqrt{15} + 2 \cdot 2}{11}$

Step 1.12.3.1.2

Multiply $15$ by $15$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{225} + \sqrt{15} \cdot 2 + 2 \sqrt{15} + 2 \cdot 2}{11}$

Step 1.12.3.1.3

Rewrite $225$ as $15^{2}$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{\sqrt{15^{2}} + \sqrt{15} \cdot 2 + 2 \sqrt{15} + 2 \cdot 2}{11}$

Step 1.12.3.1.4

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

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{15 + \sqrt{15} \cdot 2 + 2 \sqrt{15} + 2 \cdot 2}{11}$

Step 1.12.3.1.5

Move $2$ to the left of $\sqrt{15}$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{15 + 2 \cdot \sqrt{15} + 2 \sqrt{15} + 2 \cdot 2}{11}$

Step 1.12.3.1.6

Multiply $2$ by $2$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{15 + 2 \sqrt{15} + 2 \sqrt{15} + 4}{11}$

Step 1.12.3.2

Add $15$ and $4$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{19 + 2 \sqrt{15} + 2 \sqrt{15}}{11}$

Step 1.12.3.3

Add $2 \sqrt{15}$ and $2 \sqrt{15}$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} + \frac{19 + 4 \sqrt{15}}{11}$

Step 2

To write $\frac{\sqrt{255} - 2 \sqrt{17}}{17}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} \cdot \frac{11}{11} + \frac{19 + 4 \sqrt{15}}{11}$

Step 3

To write $\frac{19 + 4 \sqrt{15}}{11}$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$\frac{\sqrt{255} - 2 \sqrt{17}}{17} \cdot \frac{11}{11} + \frac{19 + 4 \sqrt{15}}{11} \cdot \frac{17}{17}$

Step 4

Write each expression with a common denominator of $187$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{\sqrt{255} - 2 \sqrt{17}}{17}$ by $\frac{11}{11}$ .

$\frac{(\sqrt{255} - 2 \sqrt{17}) \cdot 11}{17 \cdot 11} + \frac{19 + 4 \sqrt{15}}{11} \cdot \frac{17}{17}$

Step 4.2

Multiply $17$ by $11$ .

$\frac{(\sqrt{255} - 2 \sqrt{17}) \cdot 11}{187} + \frac{19 + 4 \sqrt{15}}{11} \cdot \frac{17}{17}$

Step 4.3

Multiply $\frac{19 + 4 \sqrt{15}}{11}$ by $\frac{17}{17}$ .

$\frac{(\sqrt{255} - 2 \sqrt{17}) \cdot 11}{187} + \frac{(19 + 4 \sqrt{15}) \cdot 17}{11 \cdot 17}$

Step 4.4

Multiply $11$ by $17$ .

$\frac{(\sqrt{255} - 2 \sqrt{17}) \cdot 11}{187} + \frac{(19 + 4 \sqrt{15}) \cdot 17}{187}$

Step 5

Combine the numerators over the common denominator.

$\frac{(\sqrt{255} - 2 \sqrt{17}) \cdot 11 + (19 + 4 \sqrt{15}) \cdot 17}{187}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{\sqrt{255} \cdot 11 - 2 \sqrt{17} \cdot 11 + (19 + 4 \sqrt{15}) \cdot 17}{187}$

Step 6.2

Move $11$ to the left of $\sqrt{255}$ .

$\frac{11 \cdot \sqrt{255} - 2 \sqrt{17} \cdot 11 + (19 + 4 \sqrt{15}) \cdot 17}{187}$

Step 6.3

Multiply $11$ by $- 2$ .

$\frac{11 \cdot \sqrt{255} - 22 \sqrt{17} + (19 + 4 \sqrt{15}) \cdot 17}{187}$

Step 6.4

Apply the distributive property.

$\frac{11 \sqrt{255} - 22 \sqrt{17} + 19 \cdot 17 + 4 \sqrt{15} \cdot 17}{187}$

Step 6.5

Multiply $19$ by $17$ .

$\frac{11 \sqrt{255} - 22 \sqrt{17} + 323 + 4 \sqrt{15} \cdot 17}{187}$

Step 6.6

Multiply $17$ by $4$ .

$\frac{11 \sqrt{255} - 22 \sqrt{17} + 323 + 68 \sqrt{15}}{187}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{11 \sqrt{255} - 22 \sqrt{17} + 323 + 68 \sqrt{15}}{187}$

Decimal Form:

$3.58989549 \dots$