Simplify (( square root of 6-i square root of 3)/3)(( square root of 6+i square root of 3)/3)

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

Step 1

Multiply $(\frac{\sqrt{6} - i \sqrt{3}}{3}) (\frac{\sqrt{6} + i \sqrt{3}}{3})$ .

Multiply $\frac{\sqrt{6} - i \sqrt{3}}{3}$ by $\frac{\sqrt{6} + i \sqrt{3}}{3}$ .

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

Multiply $3$ by $3$ .

$\frac{(\sqrt{6} - i \sqrt{3}) (\sqrt{6} + i \sqrt{3})}{9}$

Step 2

Expand $(\sqrt{6} - i \sqrt{3}) (\sqrt{6} + i \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{6} (\sqrt{6} + i \sqrt{3}) - i \sqrt{3} (\sqrt{6} + i \sqrt{3})}{9}$

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} (i \sqrt{3}) - i \sqrt{3} (\sqrt{6} + i \sqrt{3})}{9}$

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} (i \sqrt{3}) - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 6} + \sqrt{6} (i \sqrt{3}) - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Multiply $6$ by $6$ .

$\frac{\sqrt{36} + \sqrt{6} (i \sqrt{3}) - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2}} + \sqrt{6} (i \sqrt{3}) - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

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

$\frac{6 + \sqrt{6} (i \sqrt{3}) - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Multiply $\sqrt{6} (i \sqrt{3})$ .

Combine using the product rule for radicals.

$\frac{6 + \sqrt{3 \cdot 6} i - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Multiply $3$ by $6$ .

$\frac{6 + \sqrt{18} i - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\frac{6 + \sqrt{9 (2)} i - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

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

$\frac{6 + \sqrt{3^{2} \cdot 2} i - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Pull terms out from under the radical.

$\frac{6 + 3 \sqrt{2} i - i \sqrt{3} \sqrt{6} - i \sqrt{3} (i \sqrt{3})}{9}$

Multiply $- i \sqrt{3} \sqrt{6}$ .

Combine using the product rule for radicals.

$\frac{6 + 3 \sqrt{2} i - i \sqrt{6 \cdot 3} - i \sqrt{3} (i \sqrt{3})}{9}$

Multiply $6$ by $3$ .

$\frac{6 + 3 \sqrt{2} i - i \sqrt{18} - i \sqrt{3} (i \sqrt{3})}{9}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\frac{6 + 3 \sqrt{2} i - i \sqrt{9 (2)} - i \sqrt{3} (i \sqrt{3})}{9}$

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

$\frac{6 + 3 \sqrt{2} i - i \sqrt{3^{2} \cdot 2} - i \sqrt{3} (i \sqrt{3})}{9}$

Pull terms out from under the radical.

$\frac{6 + 3 \sqrt{2} i - i (3 \sqrt{2}) - i \sqrt{3} (i \sqrt{3})}{9}$

Multiply $3$ by $- 1$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - i \sqrt{3} (i \sqrt{3})}{9}$

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

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

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - (i^{1} i) \sqrt{3} \sqrt{3}}{9}$

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

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{9}$

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

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - i^{1 + 1} \sqrt{3} \sqrt{3}}{9}$

Add $1$ and $1$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - i^{2} \sqrt{3} \sqrt{3}}{9}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{9}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{9}$

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

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - i^{2} {\sqrt{3}}^{1 + 1}}{9}$

Add $1$ and $1$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - i^{2} {\sqrt{3}}^{2}}{9}$

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

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} - - 1 {\sqrt{3}}^{2}}{9}$

Multiply $- 1$ by $- 1$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + 1 {\sqrt{3}}^{2}}{9}$

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + {\sqrt{3}}^{2}}{9}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + {(3^{\frac{1}{2}})}^{2}}{9}$

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

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + 3^{\frac{1}{2} \cdot 2}}{9}$

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

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + 3^{\frac{2}{2}}}{9}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + 3^{\frac{2}{2}}}{9}$

Rewrite the expression.

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + 3^{1}}{9}$

Evaluate the exponent.

$\frac{6 + 3 \sqrt{2} i - 3 i \sqrt{2} + 3}{9}$

Add $6$ and $3$ .

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

Reorder the factors of $3 \sqrt{2} i$ .

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

Subtract $3 i \sqrt{2}$ from $3 i \sqrt{2}$ .

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

Add $0$ and $9$ .

$\frac{9}{9}$

Step 4

Divide $9$ by $9$ .

$1$

Simplify (( square root of 6-i square root of 3)/3)(( square root of 6+i square root of 3)/3)

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