Simplify the Radical Expression (- square root of -32)( square root of -14)

$(- \sqrt{- 32}) (\sqrt{- 14})$

Step 1

Rewrite $- 32$ as $- 1 (32)$ .

$- \sqrt{- 1 (32)} \sqrt{- 14}$

Step 2

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$- (\sqrt{- 1} \cdot \sqrt{32}) \sqrt{- 14}$

Step 3

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

$- (i \cdot \sqrt{32}) \sqrt{- 14}$

Step 4

Rewrite $- 14$ as $- 1 (14)$ .

$- (i \cdot \sqrt{32}) \sqrt{- 1 (14)}$

Step 5

Rewrite $\sqrt{- 1 (14)}$ as $\sqrt{- 1} \cdot \sqrt{14}$ .

$- (i \cdot \sqrt{32}) (\sqrt{- 1} \cdot \sqrt{14})$

Step 6

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

$- (i \cdot \sqrt{32}) (i \cdot \sqrt{14})$

Step 7

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$- (i \cdot \sqrt{16 (2)}) (i \cdot \sqrt{14})$

Rewrite $16$ as $4^{2}$ .

$- (i \cdot \sqrt{4^{2} \cdot 2}) (i \cdot \sqrt{14})$

Step 8

Pull terms out from under the radical.

$- (i \cdot (| 4 | \sqrt{2})) (i \cdot \sqrt{14})$

Step 9

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$- (i \cdot (4 \sqrt{2})) (i \cdot \sqrt{14})$

Step 10

Move $4$ to the left of $i$ .

$- (4 \cdot i \sqrt{2}) (i \cdot \sqrt{14})$

Step 11

Multiply $4$ by $- 1$ .

$- 4 (i \sqrt{2}) (i \cdot \sqrt{14})$

Step 12

Multiply $- 4 i \sqrt{2} (i \sqrt{14})$ .

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

$- 4 (i^{1} i) \sqrt{2} \sqrt{14}$

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

$- 4 (i^{1} i^{1}) \sqrt{2} \sqrt{14}$

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

$- 4 i^{1 + 1} \sqrt{2} \sqrt{14}$

Add $1$ and $1$ .

$- 4 i^{2} \sqrt{2} \sqrt{14}$

Combine using the product rule for radicals.

$- 4 i^{2} \sqrt{14 \cdot 2}$

Multiply $14$ by $2$ .

$- 4 i^{2} \sqrt{28}$

Step 13

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

$- 4 \cdot - 1 \sqrt{28}$

Step 14

Multiply $- 4$ by $- 1$ .

$4 \sqrt{28}$

Step 15

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$4 \sqrt{4 (7)}$

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

$4 \sqrt{2^{2} \cdot 7}$

Step 16

Pull terms out from under the radical.

$4 (| 2 | \sqrt{7})$

Step 17

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

$4 (2 \sqrt{7})$

Step 18

Multiply $2$ by $4$ .

$8 \sqrt{7}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt{7}$

Decimal Form:

$21.16601048 \dots$

Simplify the Radical Expression (- square root of -32)( square root of -14)

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