Algebra Examples

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

Step 1

Rewrite ${(1 + i \sqrt{3})}^{2}$ as $(1 + i \sqrt{3}) (1 + i \sqrt{3})$ .

$(1 + i \sqrt{3}) (1 + i \sqrt{3})$

Step 2

Expand $(1 + i \sqrt{3}) (1 + i \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$1 (1 + i \sqrt{3}) + i \sqrt{3} (1 + i \sqrt{3})$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1$ by $1$ .

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

Step 3.1.2

Multiply $i \sqrt{3}$ by $1$ .

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

Step 3.1.3

Multiply $i$ by $1$ .

$1 + i \sqrt{3} + i \sqrt{3} + i \sqrt{3} (i \sqrt{3})$

Step 3.1.4

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

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Step 3.1.4.1

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

$1 + i \sqrt{3} + i \sqrt{3} + i^{1} i \sqrt{3} \sqrt{3}$

Step 3.1.4.2

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

$1 + i \sqrt{3} + i \sqrt{3} + i^{1} i^{1} \sqrt{3} \sqrt{3}$

Step 3.1.4.3

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

$1 + i \sqrt{3} + i \sqrt{3} + i^{1 + 1} \sqrt{3} \sqrt{3}$

Step 3.1.4.4

Add $1$ and $1$ .

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

Step 3.1.4.5

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

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

Step 3.1.4.6

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

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

Step 3.1.4.7

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

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

Step 3.1.4.8

Add $1$ and $1$ .

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

Step 3.1.5

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

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

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

Step 3.1.6.4.2

Rewrite the expression.

$1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{1}$

Step 3.1.6.5

Evaluate the exponent.

$1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3$

Step 3.1.7

Multiply $- 1$ by $3$ .

$1 + i \sqrt{3} + i \sqrt{3} - 3$

Step 3.2

Subtract $3$ from $1$ .

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

Step 3.3

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

$2 i \sqrt{3} - 2$

Step 4

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

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