Simplify (( square root of 7-3i)/2)*(( square root of 7+3i)/4)

$(\frac{\sqrt{7} - 3 i}{2}) \cdot (\frac{\sqrt{7} + 3 i}{4})$

Multiply $(\frac{\sqrt{7} - 3 i}{2}) (\frac{\sqrt{7} + 3 i}{4})$ .

Multiply $\frac{\sqrt{7} - 3 i}{2}$ by $\frac{\sqrt{7} + 3 i}{4}$ .

$\frac{(\sqrt{7} - 3 i) (\sqrt{7} + 3 i)}{2 \cdot 4}$

Multiply $2$ by $4$ .

$\frac{(\sqrt{7} - 3 i) (\sqrt{7} + 3 i)}{8}$

Expand $(\sqrt{7} - 3 i) (\sqrt{7} + 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{7} (\sqrt{7} + 3 i) - 3 i (\sqrt{7} + 3 i)}{8}$

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{7} + \sqrt{7} (3 i) - 3 i (\sqrt{7} + 3 i)}{8}$

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{7} + \sqrt{7} (3 i) - 3 i \sqrt{7} - 3 i (3 i)}{8}$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 7} + \sqrt{7} (3 i) - 3 i \sqrt{7} - 3 i (3 i)}{8}$

Multiply $7$ by $7$ .

$\frac{\sqrt{49} + \sqrt{7} (3 i) - 3 i \sqrt{7} - 3 i (3 i)}{8}$

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{7^{2}} + \sqrt{7} (3 i) - 3 i \sqrt{7} - 3 i (3 i)}{8}$

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

$\frac{7 + \sqrt{7} (3 i) - 3 i \sqrt{7} - 3 i (3 i)}{8}$

Move $3$ to the left of $\sqrt{7}$ .

$\frac{7 + 3 \cdot \sqrt{7} i - 3 i \sqrt{7} - 3 i (3 i)}{8}$

Multiply $- 3 i (3 i)$ .

Multiply $3$ by $- 3$ .

$\frac{7 + 3 \sqrt{7} i - 3 i \sqrt{7} - 9 i i}{8}$

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

$\frac{7 + 3 \sqrt{7} i - 3 i \sqrt{7} - 9 (i^{1} i)}{8}$

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

$\frac{7 + 3 \sqrt{7} i - 3 i \sqrt{7} - 9 (i^{1} i^{1})}{8}$

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

$\frac{7 + 3 \sqrt{7} i - 3 i \sqrt{7} - 9 i^{1 + 1}}{8}$

Add $1$ and $1$ .

$\frac{7 + 3 \sqrt{7} i - 3 i \sqrt{7} - 9 i^{2}}{8}$

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

$\frac{7 + 3 \sqrt{7} i - 3 i \sqrt{7} - 9 \cdot - 1}{8}$

Multiply $- 9$ by $- 1$ .

$\frac{7 + 3 \sqrt{7} i - 3 i \sqrt{7} + 9}{8}$

Add $7$ and $9$ .

$\frac{3 \sqrt{7} i - 3 i \sqrt{7} + 16}{8}$

Reorder the factors of $3 \sqrt{7} i$ .

$\frac{3 i \sqrt{7} - 3 i \sqrt{7} + 16}{8}$

Subtract $3 i \sqrt{7}$ from $3 i \sqrt{7}$ .

$\frac{0 + 16}{8}$

Add $0$ and $16$ .

$\frac{16}{8}$

Divide $16$ by $8$ .

$2$

Simplify (( square root of 7-3i)/2)*(( square root of 7+3i)/4)

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