Trigonometry Examples

$\frac{15 - 7 \sqrt{15}}{15 + 7 \sqrt{15}}$

Step 1

Multiply $\frac{15 - 7 \sqrt{15}}{15 + 7 \sqrt{15}}$ by $\frac{15 - 7 \sqrt{15}}{15 - 7 \sqrt{15}}$ .

$\frac{15 - 7 \sqrt{15}}{15 + 7 \sqrt{15}} \cdot \frac{15 - 7 \sqrt{15}}{15 - 7 \sqrt{15}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{15 - 7 \sqrt{15}}{15 + 7 \sqrt{15}}$ by $\frac{15 - 7 \sqrt{15}}{15 - 7 \sqrt{15}}$ .

$\frac{(15 - 7 \sqrt{15}) (15 - 7 \sqrt{15})}{(15 + 7 \sqrt{15}) (15 - 7 \sqrt{15})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(15 - 7 \sqrt{15}) (15 - 7 \sqrt{15})}{225 - 105 \sqrt{15} + 105 \sqrt{15} - 49 {\sqrt{15}}^{2}}$

Step 2.3

Simplify.

$\frac{(15 - 7 \sqrt{15}) (15 - 7 \sqrt{15})}{- 510}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $15 - 7 \sqrt{15}$ to the power of $1$ .

$\frac{{(15 - 7 \sqrt{15})}^{1} (15 - 7 \sqrt{15})}{- 510}$

Step 3.2

Raise $15 - 7 \sqrt{15}$ to the power of $1$ .

$\frac{{(15 - 7 \sqrt{15})}^{1} {(15 - 7 \sqrt{15})}^{1}}{- 510}$

Step 3.3

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

$\frac{{(15 - 7 \sqrt{15})}^{1 + 1}}{- 510}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(15 - 7 \sqrt{15})}^{2}}{- 510}$

Step 4

Rewrite ${(15 - 7 \sqrt{15})}^{2}$ as $(15 - 7 \sqrt{15}) (15 - 7 \sqrt{15})$ .

$\frac{(15 - 7 \sqrt{15}) (15 - 7 \sqrt{15})}{- 510}$

Step 5

Expand $(15 - 7 \sqrt{15}) (15 - 7 \sqrt{15})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{15 (15 - 7 \sqrt{15}) - 7 \sqrt{15} (15 - 7 \sqrt{15})}{- 510}$

Step 5.2

Apply the distributive property.

$\frac{15 \cdot 15 + 15 (- 7 \sqrt{15}) - 7 \sqrt{15} (15 - 7 \sqrt{15})}{- 510}$

Step 5.3

Apply the distributive property.

$\frac{15 \cdot 15 + 15 (- 7 \sqrt{15}) - 7 \sqrt{15} \cdot 15 - 7 \sqrt{15} (- 7 \sqrt{15})}{- 510}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $15$ by $15$ .

$\frac{225 + 15 (- 7 \sqrt{15}) - 7 \sqrt{15} \cdot 15 - 7 \sqrt{15} (- 7 \sqrt{15})}{- 510}$

Step 6.1.2

Multiply $- 7$ by $15$ .

$\frac{225 - 105 \sqrt{15} - 7 \sqrt{15} \cdot 15 - 7 \sqrt{15} (- 7 \sqrt{15})}{- 510}$

Step 6.1.3

Multiply $15$ by $- 7$ .

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} - 7 \sqrt{15} (- 7 \sqrt{15})}{- 510}$

Step 6.1.4

Multiply $- 7 \sqrt{15} (- 7 \sqrt{15})$ .

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Step 6.1.4.1

Multiply $- 7$ by $- 7$ .

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 \sqrt{15} \sqrt{15}}{- 510}$

Step 6.1.4.2

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

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 ({\sqrt{15}}^{1} \sqrt{15})}{- 510}$

Step 6.1.4.3

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

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 ({\sqrt{15}}^{1} {\sqrt{15}}^{1})}{- 510}$

Step 6.1.4.4

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

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 {\sqrt{15}}^{1 + 1}}{- 510}$

Step 6.1.4.5

Add $1$ and $1$ .

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 {\sqrt{15}}^{2}}{- 510}$

Step 6.1.5

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 {(15^{\frac{1}{2}})}^{2}}{- 510}$

Step 6.1.5.2

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

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 \cdot 15^{\frac{1}{2} \cdot 2}}{- 510}$

Step 6.1.5.3

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

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 \cdot 15^{\frac{2}{2}}}{- 510}$

Step 6.1.5.4

Cancel the common factor of $2$ .

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Step 6.1.5.4.1

Cancel the common factor.

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 \cdot 15^{\frac{2}{2}}}{- 510}$

Step 6.1.5.4.2

Rewrite the expression.

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 \cdot 15^{1}}{- 510}$

Step 6.1.5.5

Evaluate the exponent.

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 49 \cdot 15}{- 510}$

Step 6.1.6

Multiply $49$ by $15$ .

$\frac{225 - 105 \sqrt{15} - 105 \sqrt{15} + 735}{- 510}$

Step 6.2

Add $225$ and $735$ .

$\frac{960 - 105 \sqrt{15} - 105 \sqrt{15}}{- 510}$

Step 6.3

Subtract $105 \sqrt{15}$ from $- 105 \sqrt{15}$ .

$\frac{960 - 210 \sqrt{15}}{- 510}$

Step 7

Cancel the common factor of $960 - 210 \sqrt{15}$ and $- 510$ .

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Step 7.1

Factor $30$ out of $960$ .

$\frac{30 (32) - 210 \sqrt{15}}{- 510}$

Step 7.2

Factor $30$ out of $- 210 \sqrt{15}$ .

$\frac{30 (32) + 30 (- 7 \sqrt{15})}{- 510}$

Step 7.3

Factor $30$ out of $30 (32) + 30 (- 7 \sqrt{15})$ .

$\frac{30 (32 - 7 \sqrt{15})}{- 510}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $30$ out of $- 510$ .

$\frac{30 (32 - 7 \sqrt{15})}{30 \cdot - 17}$

Step 7.4.2

Cancel the common factor.

$\frac{30 (32 - 7 \sqrt{15})}{30 \cdot - 17}$

Step 7.4.3

Rewrite the expression.

$\frac{32 - 7 \sqrt{15}}{- 17}$

Step 8

Move the negative in front of the fraction.

$- \frac{32 - 7 \sqrt{15}}{17}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{32 - 7 \sqrt{15}}{17}$

Decimal Form:

$- 0.28759509 \dots$