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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

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

Rewrite $4$ as $2^{2}$ .

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$2 + 2 i - i \cdot 2^{1} - i \sqrt{2} (i \sqrt{2})$

Evaluate the exponent.

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

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

$2 + 2 i - 2 i - i^{1 + 1} \sqrt{2} \sqrt{2}$

Add $1$ and $1$ .

$2 + 2 i - 2 i - i^{2} \sqrt{2} \sqrt{2}$

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

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

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

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

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

$2 + 2 i - 2 i - i^{2} {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$2 + 2 i - 2 i - i^{2} {\sqrt{2}}^{2}$

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

$2 + 2 i - 2 i - - 1 {\sqrt{2}}^{2}$

Multiply $- 1$ by $- 1$ .

$2 + 2 i - 2 i + 1 {\sqrt{2}}^{2}$

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

$2 + 2 i - 2 i + {\sqrt{2}}^{2}$

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

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

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

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

$2 + 2 i - 2 i + 2^{\frac{1}{2} \cdot 2}$

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

$2 + 2 i - 2 i + 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 + 2 i - 2 i + 2^{\frac{2}{2}}$

Rewrite the expression.

$2 + 2 i - 2 i + 2^{1}$

Evaluate the exponent.

$2 + 2 i - 2 i + 2$

Add $2$ and $2$ .

$4 + 2 i - 2 i$

Subtract $2 i$ from $2 i$ .

$4 + 0$

Add $4$ and $0$ .

$4$

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

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