Simplify square root of (3+4i)(4i-3)

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

Expand $(3 + 4 i) (4 i - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $3$ .

$\sqrt{12 i + 3 \cdot - 3 + 4 i (4 i) + 4 i \cdot - 3}$

Multiply $3$ by $- 3$ .

$\sqrt{12 i - 9 + 4 i (4 i) + 4 i \cdot - 3}$

Multiply $4 i (4 i)$ .

Multiply $4$ by $4$ .

$\sqrt{12 i - 9 + 16 i i + 4 i \cdot - 3}$

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

$\sqrt{12 i - 9 + 16 (i^{1} i) + 4 i \cdot - 3}$

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

$\sqrt{12 i - 9 + 16 (i^{1} i^{1}) + 4 i \cdot - 3}$

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

$\sqrt{12 i - 9 + 16 i^{1 + 1} + 4 i \cdot - 3}$

Add $1$ and $1$ .

$\sqrt{12 i - 9 + 16 i^{2} + 4 i \cdot - 3}$

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

$\sqrt{12 i - 9 + 16 \cdot - 1 + 4 i \cdot - 3}$

Multiply $16$ by $- 1$ .

$\sqrt{12 i - 9 - 16 + 4 i \cdot - 3}$

Multiply $- 3$ by $4$ .

$\sqrt{12 i - 9 - 16 - 12 i}$

Subtract $12 i$ from $12 i$ .

$\sqrt{0 - 9 - 16}$

Subtract $9$ from $0$ .

$\sqrt{- 9 - 16}$

Subtract $16$ from $- 9$ .

$\sqrt{- 25}$

Rewrite $- 25$ as $- 1 (25)$ .

$\sqrt{- 1 (25)}$

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$\sqrt{- 1} \cdot \sqrt{25}$

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

$i \cdot \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

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

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

$i \cdot 5$

Move $5$ to the left of $i$ .

$5 i$

Simplify square root of (3+4i)(4i-3)

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