Linear Algebra Examples

$\sqrt{- 81} \sqrt{- 144}$

Step 1

Rewrite $- 81$ as $- 1 (81)$ .

$\sqrt{- 1 (81)} \sqrt{- 144}$

Step 2

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$\sqrt{- 1} \cdot \sqrt{81} \sqrt{- 144}$

Step 3

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

$i \cdot \sqrt{81} \sqrt{- 144}$

Step 4

Rewrite $- 144$ as $- 1 (144)$ .

$i \cdot \sqrt{81} \sqrt{- 1 (144)}$

Step 5

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$i \cdot \sqrt{81} (\sqrt{- 1} \cdot \sqrt{144})$

Step 6

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

$i \cdot \sqrt{81} (i \cdot \sqrt{144})$

Step 7

Rewrite $81$ as $9^{2}$ .

$i \cdot \sqrt{9^{2}} (i \cdot \sqrt{144})$

Step 8

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

$i \cdot 9 (i \cdot \sqrt{144})$

Step 9

Move $9$ to the left of $i$ .

$9 \cdot i (i \cdot \sqrt{144})$

Step 10

Rewrite $144$ as $12^{2}$ .

$9 i (i \cdot \sqrt{12^{2}})$

Step 11

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

$9 i (i \cdot 12)$

Step 12

Move $12$ to the left of $i$ .

$9 i (12 \cdot i)$

Step 13

Multiply $9 i (12 i)$ .

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Step 13.1

Multiply $12$ by $9$ .

$108 i i$

Step 13.2

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

$108 (i^{1} i)$

Step 13.3

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

$108 (i^{1} i^{1})$

Step 13.4

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

$108 i^{1 + 1}$

Step 13.5

Add $1$ and $1$ .

$108 i^{2}$

Step 14

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

$108 \cdot - 1$

Step 15

Multiply $108$ by $- 1$ .

$- 108$