Simplify square root of (12+5i)(5i-12)

$\sqrt{(12 + 5 i) (5 i - 12)}$

Step 1

Expand $(12 + 5 i) (5 i - 12)$ using the FOIL Method.

Apply the distributive property.

$\sqrt{12 (5 i - 12) + 5 i (5 i - 12)}$

Apply the distributive property.

$\sqrt{12 (5 i) + 12 \cdot - 12 + 5 i (5 i - 12)}$

Apply the distributive property.

$\sqrt{12 (5 i) + 12 \cdot - 12 + 5 i (5 i) + 5 i \cdot - 12}$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $12$ .

$\sqrt{60 i + 12 \cdot - 12 + 5 i (5 i) + 5 i \cdot - 12}$

Multiply $12$ by $- 12$ .

$\sqrt{60 i - 144 + 5 i (5 i) + 5 i \cdot - 12}$

Multiply $5 i (5 i)$ .

Multiply $5$ by $5$ .

$\sqrt{60 i - 144 + 25 i i + 5 i \cdot - 12}$

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

$\sqrt{60 i - 144 + 25 (i^{1} i) + 5 i \cdot - 12}$

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

$\sqrt{60 i - 144 + 25 (i^{1} i^{1}) + 5 i \cdot - 12}$

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

$\sqrt{60 i - 144 + 25 i^{1 + 1} + 5 i \cdot - 12}$

Add $1$ and $1$ .

$\sqrt{60 i - 144 + 25 i^{2} + 5 i \cdot - 12}$

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

$\sqrt{60 i - 144 + 25 \cdot - 1 + 5 i \cdot - 12}$

Multiply $25$ by $- 1$ .

$\sqrt{60 i - 144 - 25 + 5 i \cdot - 12}$

Multiply $- 12$ by $5$ .

$\sqrt{60 i - 144 - 25 - 60 i}$

Subtract $60 i$ from $60 i$ .

$\sqrt{0 - 144 - 25}$

Subtract $144$ from $0$ .

$\sqrt{- 144 - 25}$

Subtract $25$ from $- 144$ .

$\sqrt{- 169}$

Step 3

Rewrite $- 169$ as $- 1 (169)$ .

$\sqrt{- 1 (169)}$

Step 4

Rewrite $\sqrt{- 1 (169)}$ as $\sqrt{- 1} \cdot \sqrt{169}$ .

$\sqrt{- 1} \cdot \sqrt{169}$

Step 5

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

$i \cdot \sqrt{169}$

Step 6

Rewrite $169$ as $13^{2}$ .

$i \cdot \sqrt{13^{2}}$

Step 7

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

$i \cdot 13$

Step 8

Move $13$ to the left of $i$ .

$13 i$

Simplify square root of (12+5i)(5i-12)

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