Algebra Examples

$\sqrt{- 20} \cdot \sqrt{- 5}$

Step 1

Rewrite $- 20$ as $- 1 (20)$ .

$\sqrt{- 1 (20)} \cdot \sqrt{- 5}$

Step 2

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$\sqrt{- 1} \cdot \sqrt{20} \cdot \sqrt{- 5}$

Step 3

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

$i \cdot \sqrt{20} \cdot \sqrt{- 5}$

Step 4

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

$i \cdot \sqrt{20} \cdot \sqrt{- 1 (5)}$

Step 5

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

$i \cdot \sqrt{20} \cdot (\sqrt{- 1} \cdot \sqrt{5})$

Step 6

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

$i \cdot \sqrt{20} \cdot (i \cdot \sqrt{5})$

Step 7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 7.1

Factor $4$ out of $20$ .

$i \cdot \sqrt{4 (5)} \cdot (i \cdot \sqrt{5})$

Step 7.2

Rewrite $4$ as $2^{2}$ .

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

Step 8

Pull terms out from under the radical.

$i \cdot (| 2 | \sqrt{5}) \cdot (i \cdot \sqrt{5})$

Step 9

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$i \cdot (2 \sqrt{5}) \cdot (i \cdot \sqrt{5})$

Step 10

Move $2$ to the left of $i$ .

$2 \cdot i \sqrt{5} \cdot (i \cdot \sqrt{5})$

Step 11

Multiply $2 i \sqrt{5} (i \sqrt{5})$ .

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Step 11.1

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

$2 (i^{1} i) \sqrt{5} \sqrt{5}$

Step 11.2

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

$2 (i^{1} i^{1}) \sqrt{5} \sqrt{5}$

Step 11.3

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

$2 i^{1 + 1} \sqrt{5} \sqrt{5}$

Step 11.4

Add $1$ and $1$ .

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

Step 11.5

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

$2 i^{2} ({\sqrt{5}}^{1} \sqrt{5})$

Step 11.6

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

$2 i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 11.7

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

$2 i^{2} {\sqrt{5}}^{1 + 1}$

Step 11.8

Add $1$ and $1$ .

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

Step 12

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

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

Step 13

Multiply $2$ by $- 1$ .

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

Cancel the common factor of $2$ .

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Step 14.4.1

Cancel the common factor.

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

Step 14.4.2

Rewrite the expression.

$- 2 \cdot 5^{1}$

Step 14.5

Evaluate the exponent.

$- 2 \cdot 5$

Step 15

Multiply $- 2$ by $5$ .

$- 10$