Simplify the Radical Expression (10 square root of 40)( square root of -9)

$(10 \sqrt{40}) (\sqrt{- 9})$

Step 1

Rewrite $- 9$ as $- 1 (9)$ .

$10 \sqrt{40} \sqrt{- 1 (9)}$

Step 2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$10 \sqrt{40} (\sqrt{- 1} \cdot \sqrt{9})$

Step 3

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

$10 \sqrt{40} (i \cdot \sqrt{9})$

Step 4

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$10 \sqrt{4 (10)} (i \cdot \sqrt{9})$

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

$10 \sqrt{2^{2} \cdot 10} (i \cdot \sqrt{9})$

Step 5

Pull terms out from under the radical.

$10 (| 2 | \sqrt{10}) (i \cdot \sqrt{9})$

Step 6

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

$10 (2 \sqrt{10}) (i \cdot \sqrt{9})$

Step 7

Multiply $2$ by $10$ .

$20 \sqrt{10} (i \cdot \sqrt{9})$

Step 8

Rewrite $9$ as $3^{2}$ .

$20 \sqrt{10} (i \cdot \sqrt{3^{2}})$

Step 9

Pull terms out from under the radical.

$20 \sqrt{10} (i \cdot | 3 |)$

Step 10

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$20 \sqrt{10} (i \cdot 3)$

Step 11

Move $3$ to the left of $i$ .

$20 \sqrt{10} (3 \cdot i)$

Step 12

Multiply $3$ by $20$ .

$60 \sqrt{10} i$