Basic Math Examples

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

Step 1

Expand $(6 - 2 i \sqrt{2}) (2 + i \sqrt{2})$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $6$ by $2$ .

$12 + 6 (i \sqrt{2}) - 2 i \sqrt{2} \cdot 2 - 2 i \sqrt{2} (i \sqrt{2})$

Step 2.1.2

Multiply $2$ by $- 2$ .

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 i \sqrt{2} (i \sqrt{2})$

Step 2.1.3

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

Tap for more steps...

Step 2.1.3.1

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 (i^{1} i) \sqrt{2} \sqrt{2}$

Step 2.1.3.2

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 (i^{1} i^{1}) \sqrt{2} \sqrt{2}$

Step 2.1.3.3

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 i^{1 + 1} \sqrt{2} \sqrt{2}$

Step 2.1.3.4

Add $1$ and $1$ .

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 i^{2} \sqrt{2} \sqrt{2}$

Step 2.1.3.5

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 i^{2} ({\sqrt{2}}^{1} \sqrt{2})$

Step 2.1.3.6

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 2.1.3.7

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 i^{2} {\sqrt{2}}^{1 + 1}$

Step 2.1.3.8

Add $1$ and $1$ .

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 i^{2} {\sqrt{2}}^{2}$

Step 2.1.4

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} - 2 \cdot - 1 {\sqrt{2}}^{2}$

Step 2.1.5

Multiply $- 2$ by $- 1$ .

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 2 {\sqrt{2}}^{2}$

Step 2.1.6

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

Tap for more steps...

Step 2.1.6.1

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 2 {(2^{\frac{1}{2}})}^{2}$

Step 2.1.6.2

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 2 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 2.1.6.3

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

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 2 \cdot 2^{\frac{2}{2}}$

Step 2.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.6.4.1

Cancel the common factor.

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 2 \cdot 2^{\frac{2}{2}}$

Step 2.1.6.4.2

Rewrite the expression.

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 2 \cdot 2^{1}$

Step 2.1.6.5

Evaluate the exponent.

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 2 \cdot 2$

Step 2.1.7

Multiply $2$ by $2$ .

$12 + 6 i \sqrt{2} - 4 i \sqrt{2} + 4$

Step 2.2

Add $12$ and $4$ .

$6 i \sqrt{2} - 4 i \sqrt{2} + 16$

Step 2.3

Subtract $4 i \sqrt{2}$ from $6 i \sqrt{2}$ .

$2 i \sqrt{2} + 16$

Step 3

Reorder $2 i \sqrt{2}$ and $16$ .

$16 + 2 i \sqrt{2}$