Simplify (i square root of -4)(i square root of -9)(i square root of -16)

$(i \sqrt{- 4}) (i \sqrt{- 9}) (i \sqrt{- 16})$

Step 1

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

$i \sqrt{- 1 (4)} (i \sqrt{- 9}) (i \sqrt{- 16})$

Step 2

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

$i (\sqrt{- 1} \cdot \sqrt{4}) (i \sqrt{- 9}) (i \sqrt{- 16})$

Step 3

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

$i (i \cdot \sqrt{4}) (i \sqrt{- 9}) (i \sqrt{- 16})$

Step 4

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

$i (i \cdot \sqrt{4}) (i \sqrt{- 1 (9)}) (i \sqrt{- 16})$

Step 5

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

$i (i \cdot \sqrt{4}) (i (\sqrt{- 1} \cdot \sqrt{9})) (i \sqrt{- 16})$

Step 6

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

$i (i \cdot \sqrt{4}) (i (i \cdot \sqrt{9})) (i \sqrt{- 16})$

Step 7

Rewrite $- 16$ as $- 1 (16)$ .

$i (i \cdot \sqrt{4}) (i (i \cdot \sqrt{9})) (i \sqrt{- 1 (16)})$

Step 8

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$i (i \cdot \sqrt{4}) (i (i \cdot \sqrt{9})) (i (\sqrt{- 1} \cdot \sqrt{16}))$

Step 9

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

$i (i \cdot \sqrt{4}) (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Step 10

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

$i (i \cdot \sqrt{2^{2}}) (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Step 11

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

$i (i \cdot 2) (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Step 12

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

$i (2 \cdot i) (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Step 13

Multiply $i (2 i)$ .

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

$2 (i^{1} i) (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

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

$2 (i^{1} i^{1}) (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

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

$2 i^{1 + 1} (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Add $1$ and $1$ .

$2 i^{2} (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Step 14

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

$2 \cdot - 1 (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Step 15

Multiply $2$ by $- 1$ .

$- 2 (i (i \cdot \sqrt{9})) (i (i \cdot \sqrt{16}))$

Step 16

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

$- 2 (i (i \cdot \sqrt{3^{2}})) (i (i \cdot \sqrt{16}))$

Step 17

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

$- 2 (i (i \cdot 3)) (i (i \cdot \sqrt{16}))$

Step 18

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

$- 2 (i (3 \cdot i)) (i (i \cdot \sqrt{16}))$

Step 19

Multiply $i (3 i)$ .

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

$- 2 (3 (i^{1} i)) (i (i \cdot \sqrt{16}))$

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

$- 2 (3 (i^{1} i^{1})) (i (i \cdot \sqrt{16}))$

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

$- 2 (3 i^{1 + 1}) (i (i \cdot \sqrt{16}))$

Add $1$ and $1$ .

$- 2 (3 i^{2}) (i (i \cdot \sqrt{16}))$

Step 20

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

$- 2 (3 \cdot - 1) (i (i \cdot \sqrt{16}))$

Step 21

Multiply $- 2 (3 \cdot - 1)$ .

Multiply $3$ by $- 1$ .

$- 2 \cdot - 3 (i (i \cdot \sqrt{16}))$

Multiply $- 2$ by $- 3$ .

$6 (i (i \cdot \sqrt{16}))$

Step 22

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

$6 (i (i \cdot \sqrt{4^{2}}))$

Step 23

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

$6 (i (i \cdot 4))$

Step 24

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

$6 (i (4 \cdot i))$

Step 25

Multiply $i (4 i)$ .

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

$6 (4 (i^{1} i))$

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

$6 (4 (i^{1} i^{1}))$

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

$6 (4 i^{1 + 1})$

Add $1$ and $1$ .

$6 (4 i^{2})$

Step 26

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

$6 (4 \cdot - 1)$

Step 27

Multiply $6 (4 \cdot - 1)$ .

Multiply $4$ by $- 1$ .

$6 \cdot - 4$

Multiply $6$ by $- 4$ .

$- 24$

Simplify (i square root of -4)(i square root of -9)(i square root of -16)

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