Algebra Examples

$\sqrt{8} \cdot \sqrt{- 8}$

Step 1

Rewrite $- 8$ as $- 1 (8)$ .

$\sqrt{8} \cdot \sqrt{- 1 (8)}$

Step 2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\sqrt{8} \cdot (\sqrt{- 1} \cdot \sqrt{8})$

Step 3

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

$\sqrt{8} \cdot (i \cdot \sqrt{8})$

Step 4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)} \cdot (i \cdot \sqrt{8})$

Step 4.2

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

$\sqrt{2^{2} \cdot 2} \cdot (i \cdot \sqrt{8})$

Step 5

Pull terms out from under the radical.

$2 \sqrt{2} \cdot (i \cdot \sqrt{8})$

Step 6

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.1

Factor $4$ out of $8$ .

$2 \sqrt{2} \cdot (i \cdot \sqrt{4 (2)})$

Step 6.2

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

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

Step 7

Pull terms out from under the radical.

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

Step 8

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

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

Step 9

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

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Step 9.1

Multiply $2$ by $2$ .

$4 \sqrt{2} (i \sqrt{2})$

Step 9.2

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

$4 ({\sqrt{2}}^{1} \sqrt{2}) i$

Step 9.3

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

$4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) i$

Step 9.4

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

$4 {\sqrt{2}}^{1 + 1} i$

Step 9.5

Add $1$ and $1$ .

$4 {\sqrt{2}}^{2} i$

Step 10

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

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Step 10.1

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

$4 {(2^{\frac{1}{2}})}^{2} i$

Step 10.2

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

$4 \cdot 2^{\frac{1}{2} \cdot 2} i$

Step 10.3

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

$4 \cdot 2^{\frac{2}{2}} i$

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

$4 \cdot 2^{\frac{2}{2}} i$

Step 10.4.2

Rewrite the expression.

$4 \cdot 2^{1} i$

Step 10.5

Evaluate the exponent.

$4 \cdot 2 i$

Step 11

Multiply $4$ by $2$ .

$8 i$