Simplify (( square root of 5-2i)/2)(( square root of 5+2i)/2)

$(\frac{\sqrt{5} - 2 i}{2}) (\frac{\sqrt{5} + 2 i}{2})$

Multiply $(\frac{\sqrt{5} - 2 i}{2}) (\frac{\sqrt{5} + 2 i}{2})$ .

Multiply $\frac{\sqrt{5} - 2 i}{2}$ by $\frac{\sqrt{5} + 2 i}{2}$ .

$\frac{(\sqrt{5} - 2 i) (\sqrt{5} + 2 i)}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{(\sqrt{5} - 2 i) (\sqrt{5} + 2 i)}{4}$

Expand $(\sqrt{5} - 2 i) (\sqrt{5} + 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{5} (\sqrt{5} + 2 i) - 2 i (\sqrt{5} + 2 i)}{4}$

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{5} + \sqrt{5} (2 i) - 2 i (\sqrt{5} + 2 i)}{4}$

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{5} + \sqrt{5} (2 i) - 2 i \sqrt{5} - 2 i (2 i)}{4}$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 5} + \sqrt{5} (2 i) - 2 i \sqrt{5} - 2 i (2 i)}{4}$

Multiply $5$ by $5$ .

$\frac{\sqrt{25} + \sqrt{5} (2 i) - 2 i \sqrt{5} - 2 i (2 i)}{4}$

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

$\frac{\sqrt{5^{2}} + \sqrt{5} (2 i) - 2 i \sqrt{5} - 2 i (2 i)}{4}$

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

$\frac{5 + \sqrt{5} (2 i) - 2 i \sqrt{5} - 2 i (2 i)}{4}$

Move $2$ to the left of $\sqrt{5}$ .

$\frac{5 + 2 \cdot \sqrt{5} i - 2 i \sqrt{5} - 2 i (2 i)}{4}$

Multiply $- 2 i (2 i)$ .

Multiply $2$ by $- 2$ .

$\frac{5 + 2 \sqrt{5} i - 2 i \sqrt{5} - 4 i i}{4}$

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

$\frac{5 + 2 \sqrt{5} i - 2 i \sqrt{5} - 4 (i^{1} i)}{4}$

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

$\frac{5 + 2 \sqrt{5} i - 2 i \sqrt{5} - 4 (i^{1} i^{1})}{4}$

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

$\frac{5 + 2 \sqrt{5} i - 2 i \sqrt{5} - 4 i^{1 + 1}}{4}$

Add $1$ and $1$ .

$\frac{5 + 2 \sqrt{5} i - 2 i \sqrt{5} - 4 i^{2}}{4}$

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

$\frac{5 + 2 \sqrt{5} i - 2 i \sqrt{5} - 4 \cdot - 1}{4}$

Multiply $- 4$ by $- 1$ .

$\frac{5 + 2 \sqrt{5} i - 2 i \sqrt{5} + 4}{4}$

Add $5$ and $4$ .

$\frac{2 \sqrt{5} i - 2 i \sqrt{5} + 9}{4}$

Reorder the factors of $2 \sqrt{5} i$ .

$\frac{2 i \sqrt{5} - 2 i \sqrt{5} + 9}{4}$

Subtract $2 i \sqrt{5}$ from $2 i \sqrt{5}$ .

$\frac{0 + 9}{4}$

Add $0$ and $9$ .

$\frac{9}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{4}$

Decimal Form:

$2.25$

Mixed Number Form:

$2 \frac{1}{4}$

Simplify (( square root of 5-2i)/2)(( square root of 5+2i)/2)

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