Algebra Examples

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

Step 1

Expand $(i \sqrt{6} + 3) (i \sqrt{6} + 2)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Add $1$ and $1$ .

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

Step 2.1.1.5

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 2.1.1.6

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 2.1.1.7

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

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

Step 2.1.1.8

Add $1$ and $1$ .

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

Step 2.1.2

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

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

Step 2.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Cancel the common factor.

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

Step 2.1.3.4.2

Rewrite the expression.

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

Step 2.1.3.5

Evaluate the exponent.

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

Step 2.1.4

Multiply $- 1$ by $6$ .

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

Step 2.1.5

Move $2$ to the left of $i \sqrt{6}$ .

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

Step 2.1.6

Multiply $3$ by $2$ .

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

Step 2.2

Add $- 6$ and $6$ .

$2 i \sqrt{6} + 3 i \sqrt{6} + 0$

Step 2.3

Add $2 i \sqrt{6}$ and $3 i \sqrt{6}$ .

$5 i \sqrt{6} + 0$

Step 2.4

Add $5 i \sqrt{6}$ and $0$ .

$5 i \sqrt{6}$