Algebra Examples

$(\sqrt{- 4} + \sqrt{- 9}) (\sqrt{- 4} - \sqrt{- 9})$

Step 1

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

$(\sqrt{- 1 (4)} + \sqrt{- 9}) (\sqrt{- 4} - \sqrt{- 9})$

Step 2

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

$(\sqrt{- 1} \cdot \sqrt{4} + \sqrt{- 9}) (\sqrt{- 4} - \sqrt{- 9})$

Step 3

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

$(i \cdot \sqrt{4} + \sqrt{- 9}) (\sqrt{- 4} - \sqrt{- 9})$

Step 4

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

$(i \cdot \sqrt{4} + \sqrt{- 1 (9)}) (\sqrt{- 4} - \sqrt{- 9})$

Step 5

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

$(i \cdot \sqrt{4} + \sqrt{- 1} \cdot \sqrt{9}) (\sqrt{- 4} - \sqrt{- 9})$

Step 6

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

$(i \cdot \sqrt{4} + i \cdot \sqrt{9}) (\sqrt{- 4} - \sqrt{- 9})$

Step 7

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

$(i \cdot \sqrt{4} + i \cdot \sqrt{9}) (\sqrt{- 1 (4)} - \sqrt{- 9})$

Step 8

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

$(i \cdot \sqrt{4} + i \cdot \sqrt{9}) (\sqrt{- 1} \cdot \sqrt{4} - \sqrt{- 9})$

Step 9

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

$(i \cdot \sqrt{4} + i \cdot \sqrt{9}) (i \cdot \sqrt{4} - \sqrt{- 9})$

Step 10

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

$(i \cdot \sqrt{4} + i \cdot \sqrt{9}) (i \cdot \sqrt{4} - \sqrt{- 1 (9)})$

Step 11

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

$(i \cdot \sqrt{4} + i \cdot \sqrt{9}) (i \cdot \sqrt{4} - (\sqrt{- 1} \cdot \sqrt{9}))$

Step 12

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

$(i \cdot \sqrt{4} + i \cdot \sqrt{9}) (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 13

Simplify each term.

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Step 13.1

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

$(i \cdot \sqrt{2^{2}} + i \cdot \sqrt{9}) (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 13.2

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

$(i \cdot 2 + i \cdot \sqrt{9}) (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 13.3

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

$(2 \cdot i + i \cdot \sqrt{9}) (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 13.4

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

$(2 i + i \cdot \sqrt{3^{2}}) (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 13.5

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

$(2 i + i \cdot 3) (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 13.6

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

$(2 i + 3 i) (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 14

Add $2 i$ and $3 i$ .

$5 i (i \cdot \sqrt{4} - (i \cdot \sqrt{9}))$

Step 15

Simplify each term.

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Step 15.1

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

$5 i (i \cdot \sqrt{2^{2}} - (i \cdot \sqrt{9}))$

Step 15.2

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

$5 i (i \cdot 2 - (i \cdot \sqrt{9}))$

Step 15.3

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

$5 i (2 \cdot i - (i \cdot \sqrt{9}))$

Step 15.4

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

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

Step 15.5

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

$5 i (2 i - (i \cdot 3))$

Step 15.6

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

$5 i (2 i - (3 \cdot i))$

Step 15.7

Multiply $3$ by $- 1$ .

$5 i (2 i - 3 i)$

Step 16

Subtract $3 i$ from $2 i$ .

$5 i (- i)$

Step 17

Multiply $5 i (- i)$ .

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Step 17.1

Multiply $- 1$ by $5$ .

$- 5 i i$

Step 17.2

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

$- 5 (i^{1} i)$

Step 17.3

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

$- 5 (i^{1} i^{1})$

Step 17.4

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

$- 5 i^{1 + 1}$

Step 17.5

Add $1$ and $1$ .

$- 5 i^{2}$

Step 18

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

$- 5 \cdot - 1$

Step 19

Multiply $- 5$ by $- 1$ .

$5$