Basic Math Examples

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

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

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

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

Step 1.2

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

$\frac{\sqrt{- 1} \cdot \sqrt{15} \cdot \sqrt{- 5}}{\sqrt{3}}$

Step 1.3

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

$\frac{i \cdot \sqrt{15} \cdot \sqrt{- 5}}{\sqrt{3}}$

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 2

Combine $\sqrt{15}$ and $\sqrt{3}$ into a single radical.

$i \cdot \sqrt{\frac{15}{3}} \cdot i \cdot \sqrt{5}$

Step 3

Divide $15$ by $3$ .

$i \cdot \sqrt{5} \cdot i \cdot \sqrt{5}$

Step 4

Multiply $i \sqrt{5} i$ .

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Step 4.1

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

$i^{1} i \sqrt{5} \cdot \sqrt{5}$

Step 4.2

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

$i^{1} i^{1} \sqrt{5} \cdot \sqrt{5}$

Step 4.3

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

$i^{1 + 1} \sqrt{5} \cdot \sqrt{5}$

Step 4.4

Add $1$ and $1$ .

$i^{2} \sqrt{5} \cdot \sqrt{5}$

Step 5

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

$- 1 \sqrt{5} \cdot \sqrt{5}$

Step 6

Rewrite $- 1 \sqrt{5}$ as $- \sqrt{5}$ .

$- \sqrt{5} \cdot \sqrt{5}$

Step 7

Multiply $- \sqrt{5} \sqrt{5}$ .

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Step 7.1

Raise $\sqrt{5}$ to the power of $1$ .

$- ({\sqrt{5}}^{1} \sqrt{5})$

Step 7.2

Raise $\sqrt{5}$ to the power of $1$ .

$- ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 7.3

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

$- {\sqrt{5}}^{1 + 1}$

Step 7.4

Add $1$ and $1$ .

$- {\sqrt{5}}^{2}$

Step 8

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$- {(5^{\frac{1}{2}})}^{2}$

Step 8.2

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

$- 5^{\frac{1}{2} \cdot 2}$

Step 8.3

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

$- 5^{\frac{2}{2}}$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

$- 5^{\frac{2}{2}}$

Step 8.4.2

Rewrite the expression.

$- 5^{1}$

Step 8.5

Evaluate the exponent.

$- 1 \cdot 5$

Step 9

Multiply $- 1$ by $5$ .

$- 5$