Pre-Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2$ by $3$ .

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

Step 2.1.2

Multiply $- 1$ by $2$ .

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

Step 2.1.3

Move $3$ to the left of $\sqrt{2 i}$ .

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

Step 2.1.4

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

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

Add $1$ and $1$ .

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

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

Step 2.1.5.4.2

Rewrite the expression.

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

Step 2.1.5.5

Simplify.

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

Step 2.1.6

Multiply $2$ by $- 1$ .

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

Step 2.2

Add $- 2 \sqrt{2 i}$ and $3 \sqrt{2 i}$ .

$6 + \sqrt{2 i} - 2 i$