Simplify i square root of -45

$i \sqrt{- 45}$

Step 1

Rewrite $- 45$ as $- 1 (45)$ .

$i \sqrt{- 1 (45)}$

Step 2

Rewrite $\sqrt{- 1 (45)}$ as $\sqrt{- 1} \cdot \sqrt{45}$ .

$i (\sqrt{- 1} \cdot \sqrt{45})$

Step 3

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

$i (i \cdot \sqrt{45})$

Step 4

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$i (i \cdot \sqrt{9 (5)})$

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

$i (i \cdot \sqrt{3^{2} \cdot 5})$

Step 5

Pull terms out from under the radical.

$i (i \cdot (3 \sqrt{5}))$

Step 6

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

$i (3 \cdot i \sqrt{5})$

Step 7

Multiply $i (3 i \sqrt{5})$ .

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

$3 (i^{1} i) \sqrt{5}$

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

$3 (i^{1} i^{1}) \sqrt{5}$

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

$3 i^{1 + 1} \sqrt{5}$

Add $1$ and $1$ .

$3 i^{2} \sqrt{5}$

Step 8

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

$3 \cdot - 1 \sqrt{5}$

Step 9

Multiply $3$ by $- 1$ .

$- 3 \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 3 \sqrt{5}$

Decimal Form:

$- 6.70820393 \dots$