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

$\frac{\sqrt{3} + i}{\sqrt{3} - i}$

Step 1

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

$\frac{\sqrt{3} + i}{\sqrt{3} - i} \cdot \frac{\sqrt{3} + i}{\sqrt{3} + i}$

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{3} + \sqrt{3} i + i \sqrt{3} + i i}{4}$

Step 4

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 3} + \sqrt{3} i + i \sqrt{3} + i i}{4}$

Multiply $3$ by $3$ .

$\frac{\sqrt{9} + \sqrt{3} i + i \sqrt{3} + i i}{4}$

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} + \sqrt{3} i + i \sqrt{3} + i i}{4}$

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

$\frac{3 + \sqrt{3} i + i \sqrt{3} + i i}{4}$

Multiply $i i$ .

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

$\frac{3 + \sqrt{3} i + i \sqrt{3} + i^{1} i}{4}$

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

$\frac{3 + \sqrt{3} i + i \sqrt{3} + i^{1} i^{1}}{4}$

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

$\frac{3 + \sqrt{3} i + i \sqrt{3} + i^{1 + 1}}{4}$

Add $1$ and $1$ .

$\frac{3 + \sqrt{3} i + i \sqrt{3} + i^{2}}{4}$

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

$\frac{3 + \sqrt{3} i + i \sqrt{3} - 1}{4}$

Subtract $1$ from $3$ .

$\frac{2 + \sqrt{3} i + i \sqrt{3}}{4}$

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

$\frac{2 + i \sqrt{3} + i \sqrt{3}}{4}$

Add $i \sqrt{3}$ and $i \sqrt{3}$ .

$\frac{2 + 2 i \sqrt{3}}{4}$

Step 5

Cancel the common factor of $2 + 2 i \sqrt{3}$ and $4$ .

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 + 2 i \sqrt{3}}{4}$

Factor $2$ out of $2 i \sqrt{3}$ .

$\frac{2 \cdot 1 + 2 (i \sqrt{3})}{4}$

Factor $2$ out of $2 \cdot 1 + 2 (i \sqrt{3})$ .

$\frac{2 \cdot (1 + i \sqrt{3})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 \cdot (1 + i \sqrt{3})}{2 (2)}$

Cancel the common factor.

$\frac{2 \cdot (1 + i \sqrt{3})}{2 \cdot 2}$

Rewrite the expression.

$\frac{1 + i \sqrt{3}}{2}$

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

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