Trigonometry Examples

$- \sqrt{- 5} \sqrt{- 3} \sqrt{- 15}$

Step 1

Rewrite $- 5$ as $- 1 (5)$ .

$- \sqrt{- 1 (5)} \sqrt{- 3} \sqrt{- 15}$

Step 2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$- (\sqrt{- 1} \cdot \sqrt{5}) \sqrt{- 3} \sqrt{- 15}$

Step 3

Rewrite $\sqrt{- 1}$ as $i$ .

$- (i \cdot \sqrt{5}) \sqrt{- 3} \sqrt{- 15}$

Step 4

Rewrite $- 3$ as $- 1 (3)$ .

$- (i \cdot \sqrt{5}) \sqrt{- 1 (3)} \sqrt{- 15}$

Step 5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$- (i \cdot \sqrt{5}) (\sqrt{- 1} \cdot \sqrt{3}) \sqrt{- 15}$

Step 6

Rewrite $\sqrt{- 1}$ as $i$ .

$- (i \cdot \sqrt{5}) (i \cdot \sqrt{3}) \sqrt{- 15}$

Step 7

Rewrite $- 15$ as $- 1 (15)$ .

$- (i \cdot \sqrt{5}) (i \cdot \sqrt{3}) \sqrt{- 1 (15)}$

Step 8

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$- (i \cdot \sqrt{5}) (i \cdot \sqrt{3}) (\sqrt{- 1} \cdot \sqrt{15})$

Step 9

Rewrite $\sqrt{- 1}$ as $i$ .

$- (i \cdot \sqrt{5}) (i \cdot \sqrt{3}) (i \cdot \sqrt{15})$

Step 10

Multiply $- i \sqrt{5} (i \sqrt{3})$ .

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Step 10.1

Raise $i$ to the power of $1$ .

$- (i^{1} i) \sqrt{5} \sqrt{3} (i \cdot \sqrt{15})$

Step 10.2

Raise $i$ to the power of $1$ .

$- (i^{1} i^{1}) \sqrt{5} \sqrt{3} (i \cdot \sqrt{15})$

Step 10.3

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

$- i^{1 + 1} \sqrt{5} \sqrt{3} (i \cdot \sqrt{15})$

Step 10.4

Add $1$ and $1$ .

$- i^{2} \sqrt{5} \sqrt{3} (i \cdot \sqrt{15})$

Step 10.5

Combine using the product rule for radicals.

$- i^{2} \sqrt{3 \cdot 5} (i \cdot \sqrt{15})$

Step 10.6

Multiply $3$ by $5$ .

$- i^{2} \sqrt{15} (i \cdot \sqrt{15})$

Step 11

Rewrite $i^{2}$ as $- 1$ .

$- - 1 \sqrt{15} (i \cdot \sqrt{15})$

Step 12

Multiply $- 1$ by $- 1$ .

$1 \sqrt{15} (i \cdot \sqrt{15})$

Step 13

Multiply $\sqrt{15}$ by $1$ .

$\sqrt{15} (i \cdot \sqrt{15})$

Step 14

Multiply $\sqrt{15} (i \sqrt{15})$ .

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Step 14.1

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

${\sqrt{15}}^{1} \sqrt{15} i$

Step 14.2

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

${\sqrt{15}}^{1} {\sqrt{15}}^{1} i$

Step 14.3

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

${\sqrt{15}}^{1 + 1} i$

Step 14.4

Add $1$ and $1$ .

${\sqrt{15}}^{2} i$

Step 15

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

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Step 15.1

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

${(15^{\frac{1}{2}})}^{2} i$

Step 15.2

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

$15^{\frac{1}{2} \cdot 2} i$

Step 15.3

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

$15^{\frac{2}{2}} i$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Cancel the common factor.

$15^{\frac{2}{2}} i$

Step 15.4.2

Rewrite the expression.

$15^{1} i$

Step 15.5

Evaluate the exponent.

$15 i$