Simplify square root of (6+8i)(8i-6)

$\sqrt{(6 + 8 i) (8 i - 6)}$

Step 1

Expand $(6 + 8 i) (8 i - 6)$ using the FOIL Method.

Apply the distributive property.

$\sqrt{6 (8 i - 6) + 8 i (8 i - 6)}$

Apply the distributive property.

$\sqrt{6 (8 i) + 6 \cdot - 6 + 8 i (8 i - 6)}$

Apply the distributive property.

$\sqrt{6 (8 i) + 6 \cdot - 6 + 8 i (8 i) + 8 i \cdot - 6}$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $6$ .

$\sqrt{48 i + 6 \cdot - 6 + 8 i (8 i) + 8 i \cdot - 6}$

Multiply $6$ by $- 6$ .

$\sqrt{48 i - 36 + 8 i (8 i) + 8 i \cdot - 6}$

Multiply $8 i (8 i)$ .

Multiply $8$ by $8$ .

$\sqrt{48 i - 36 + 64 i i + 8 i \cdot - 6}$

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

$\sqrt{48 i - 36 + 64 (i^{1} i) + 8 i \cdot - 6}$

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

$\sqrt{48 i - 36 + 64 (i^{1} i^{1}) + 8 i \cdot - 6}$

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

$\sqrt{48 i - 36 + 64 i^{1 + 1} + 8 i \cdot - 6}$

Add $1$ and $1$ .

$\sqrt{48 i - 36 + 64 i^{2} + 8 i \cdot - 6}$

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

$\sqrt{48 i - 36 + 64 \cdot - 1 + 8 i \cdot - 6}$

Multiply $64$ by $- 1$ .

$\sqrt{48 i - 36 - 64 + 8 i \cdot - 6}$

Multiply $- 6$ by $8$ .

$\sqrt{48 i - 36 - 64 - 48 i}$

Subtract $48 i$ from $48 i$ .

$\sqrt{0 - 36 - 64}$

Subtract $36$ from $0$ .

$\sqrt{- 36 - 64}$

Subtract $64$ from $- 36$ .

$\sqrt{- 100}$

Step 3

Rewrite $- 100$ as $- 1 (100)$ .

$\sqrt{- 1 (100)}$

Step 4

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$\sqrt{- 1} \cdot \sqrt{100}$

Step 5

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

$i \cdot \sqrt{100}$

Step 6

Rewrite $100$ as $10^{2}$ .

$i \cdot \sqrt{10^{2}}$

Step 7

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

$i \cdot 10$

Step 8

Move $10$ to the left of $i$ .

$10 i$

Simplify square root of (6+8i)(8i-6)

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