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

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

Step 1

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

$\frac{\sqrt{3} + i \sqrt{2}}{\sqrt{3} - i \sqrt{2}} \cdot \frac{\sqrt{3} + i \sqrt{2}}{\sqrt{3} + i \sqrt{2}}$

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} + i \sqrt{2}) (\sqrt{3} + i \sqrt{2})}{{\sqrt{3}}^{2} + \sqrt{6} i - i \sqrt{6} - i^{2} {\sqrt{2}}^{2}}$

Simplify.

$\frac{(\sqrt{3} + i \sqrt{2}) (\sqrt{3} + i \sqrt{2})}{5}$

Step 3

Expand $(\sqrt{3} + i \sqrt{2}) (\sqrt{3} + i \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{3} (\sqrt{3} + i \sqrt{2}) + i \sqrt{2} (\sqrt{3} + i \sqrt{2})}{5}$

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{3} + \sqrt{3} (i \sqrt{2}) + i \sqrt{2} (\sqrt{3} + i \sqrt{2})}{5}$

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{3} + \sqrt{3} (i \sqrt{2}) + i \sqrt{2} \sqrt{3} + i \sqrt{2} (i \sqrt{2})}{5}$

Step 4

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 3} + \sqrt{3} (i \sqrt{2}) + i \sqrt{2} \sqrt{3} + i \sqrt{2} (i \sqrt{2})}{5}$

Multiply $3$ by $3$ .

$\frac{\sqrt{9} + \sqrt{3} (i \sqrt{2}) + i \sqrt{2} \sqrt{3} + i \sqrt{2} (i \sqrt{2})}{5}$

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

$\frac{\sqrt{3^{2}} + \sqrt{3} (i \sqrt{2}) + i \sqrt{2} \sqrt{3} + i \sqrt{2} (i \sqrt{2})}{5}$

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

$\frac{3 + \sqrt{3} (i \sqrt{2}) + i \sqrt{2} \sqrt{3} + i \sqrt{2} (i \sqrt{2})}{5}$

Multiply $\sqrt{3} (i \sqrt{2})$ .

Combine using the product rule for radicals.

$\frac{3 + \sqrt{2 \cdot 3} i + i \sqrt{2} \sqrt{3} + i \sqrt{2} (i \sqrt{2})}{5}$

Multiply $2$ by $3$ .

$\frac{3 + \sqrt{6} i + i \sqrt{2} \sqrt{3} + i \sqrt{2} (i \sqrt{2})}{5}$

Multiply $i \sqrt{2} \sqrt{3}$ .

Combine using the product rule for radicals.

$\frac{3 + \sqrt{6} i + i \sqrt{3 \cdot 2} + i \sqrt{2} (i \sqrt{2})}{5}$

Multiply $3$ by $2$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i \sqrt{2} (i \sqrt{2})}{5}$

Multiply $i \sqrt{2} (i \sqrt{2})$ .

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

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{1} i \sqrt{2} \sqrt{2}}{5}$

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

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{1} i^{1} \sqrt{2} \sqrt{2}}{5}$

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

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{1 + 1} \sqrt{2} \sqrt{2}}{5}$

Add $1$ and $1$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{2} \sqrt{2} \sqrt{2}}{5}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{2} ({\sqrt{2}}^{1} \sqrt{2})}{5}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{5}$

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

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{2} {\sqrt{2}}^{1 + 1}}{5}$

Add $1$ and $1$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} + i^{2} {\sqrt{2}}^{2}}{5}$

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

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 1 {\sqrt{2}}^{2}}{5}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 1 {(2^{\frac{1}{2}})}^{2}}{5}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 1 \cdot 2^{\frac{1}{2} \cdot 2}}{5}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 1 \cdot 2^{\frac{2}{2}}}{5}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 1 \cdot 2^{\frac{2}{2}}}{5}$

Rewrite the expression.

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 1 \cdot 2^{1}}{5}$

Evaluate the exponent.

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 1 \cdot 2}{5}$

Multiply $- 1$ by $2$ .

$\frac{3 + \sqrt{6} i + i \sqrt{6} - 2}{5}$

Subtract $2$ from $3$ .

$\frac{1 + \sqrt{6} i + i \sqrt{6}}{5}$

Reorder the factors of $\sqrt{6} i$ .

$\frac{1 + i \sqrt{6} + i \sqrt{6}}{5}$

Add $i \sqrt{6}$ and $i \sqrt{6}$ .

$\frac{1 + 2 i \sqrt{6}}{5}$

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

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