Algebra Examples

$\sqrt{- 4} \cdot \sqrt{2} \cdot \sqrt{- 8} \cdot \sqrt{4}$

Step 1

Rewrite $- 4$ as $- 1 (4)$ .

$\sqrt{- 1 (4)} \cdot \sqrt{2} \cdot \sqrt{- 8} \cdot \sqrt{4}$

Step 2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$\sqrt{- 1} \cdot \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{- 8} \cdot \sqrt{4}$

Step 3

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

$i \cdot \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{- 8} \cdot \sqrt{4}$

Step 4

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

$i \cdot \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{- 1 (8)} \cdot \sqrt{4}$

Step 5

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

$i \cdot \sqrt{4} \cdot \sqrt{2} \cdot (\sqrt{- 1} \cdot \sqrt{8}) \cdot \sqrt{4}$

Step 6

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

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

Step 7

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

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

Step 8

Pull terms out from under the radical, assuming positive real numbers.

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

Step 9

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

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

Step 10

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

Tap for more steps...

Step 10.1

Factor $4$ out of $8$ .

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

Step 10.2

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

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

Step 11

Pull terms out from under the radical.

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

Step 12

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

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

Step 13

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

Tap for more steps...

Step 13.1

Multiply $2$ by $2$ .

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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

Step 13.5

Add $1$ and $1$ .

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

Step 13.6

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

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

Step 13.7

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

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

Step 13.8

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

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

Step 13.9

Add $1$ and $1$ .

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

Step 14

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

$4 \cdot - 1 {\sqrt{2}}^{2} \cdot \sqrt{4}$

Step 15

Multiply $4$ by $- 1$ .

$- 4 {\sqrt{2}}^{2} \cdot \sqrt{4}$

Step 16

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

Tap for more steps...

Step 16.1

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

$- 4 {(2^{\frac{1}{2}})}^{2} \cdot \sqrt{4}$

Step 16.2

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

$- 4 \cdot 2^{\frac{1}{2} \cdot 2} \cdot \sqrt{4}$

Step 16.3

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

$- 4 \cdot 2^{\frac{2}{2}} \cdot \sqrt{4}$

Step 16.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 16.4.1

Cancel the common factor.

$- 4 \cdot 2^{\frac{2}{2}} \cdot \sqrt{4}$

Step 16.4.2

Rewrite the expression.

$- 4 \cdot 2^{1} \cdot \sqrt{4}$

Step 16.5

Evaluate the exponent.

$- 4 \cdot 2 \cdot \sqrt{4}$

Step 17

Write the expression using exponents.

Tap for more steps...

Step 17.1

Multiply $- 4$ by $2$ .

$- 8 \cdot \sqrt{4}$

Step 17.2

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

$- 8 \cdot \sqrt{2^{2}}$

Step 18

Pull terms out from under the radical, assuming positive real numbers.

$- 8 \cdot 2$

Step 19

Multiply $- 8$ by $2$ .

$- 16$