Precalculus Examples

${(- 4 + 4 i \sqrt{3})}^{2}$

Step 1

Rewrite ${(- 4 + 4 i \sqrt{3})}^{2}$ as $(- 4 + 4 i \sqrt{3}) (- 4 + 4 i \sqrt{3})$ .

$(- 4 + 4 i \sqrt{3}) (- 4 + 4 i \sqrt{3})$

Step 2

Expand $(- 4 + 4 i \sqrt{3}) (- 4 + 4 i \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$- 4 (- 4 + 4 i \sqrt{3}) + 4 i \sqrt{3} (- 4 + 4 i \sqrt{3})$

Step 2.2

Apply the distributive property.

$- 4 \cdot - 4 - 4 (4 i \sqrt{3}) + 4 i \sqrt{3} (- 4 + 4 i \sqrt{3})$

Step 2.3

Apply the distributive property.

$- 4 \cdot - 4 - 4 (4 i \sqrt{3}) + 4 i \sqrt{3} \cdot - 4 + 4 i \sqrt{3} (4 i \sqrt{3})$

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 $- 4$ by $- 4$ .

$16 - 4 (4 i \sqrt{3}) + 4 i \sqrt{3} \cdot - 4 + 4 i \sqrt{3} (4 i \sqrt{3})$

Step 3.1.2

Multiply $4$ by $- 4$ .

$16 - 16 (i \sqrt{3}) + 4 i \sqrt{3} \cdot - 4 + 4 i \sqrt{3} (4 i \sqrt{3})$

Step 3.1.3

Multiply $- 4$ by $4$ .

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 4 i \sqrt{3} (4 i \sqrt{3})$

Step 3.1.4

Multiply $4 i \sqrt{3} (4 i \sqrt{3})$ .

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Step 3.1.4.1

Multiply $4$ by $4$ .

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 i \sqrt{3} (i \sqrt{3})$

Step 3.1.4.2

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 (i^{1} i) \sqrt{3} \sqrt{3}$

Step 3.1.4.3

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 (i^{1} i^{1}) \sqrt{3} \sqrt{3}$

Step 3.1.4.4

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 i^{1 + 1} \sqrt{3} \sqrt{3}$

Step 3.1.4.5

Add $1$ and $1$ .

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 i^{2} \sqrt{3} \sqrt{3}$

Step 3.1.4.6

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 3.1.4.7

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 3.1.4.8

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 i^{2} {\sqrt{3}}^{1 + 1}$

Step 3.1.4.9

Add $1$ and $1$ .

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 i^{2} {\sqrt{3}}^{2}$

Step 3.1.5

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} + 16 \cdot - 1 {\sqrt{3}}^{2}$

Step 3.1.6

Multiply $16$ by $- 1$ .

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 16 {\sqrt{3}}^{2}$

Step 3.1.7

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

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Step 3.1.7.1

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 16 {(3^{\frac{1}{2}})}^{2}$

Step 3.1.7.2

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 16 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 3.1.7.3

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

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 16 \cdot 3^{\frac{2}{2}}$

Step 3.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.7.4.1

Cancel the common factor.

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 16 \cdot 3^{\frac{2}{2}}$

Step 3.1.7.4.2

Rewrite the expression.

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 16 \cdot 3^{1}$

Step 3.1.7.5

Evaluate the exponent.

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 16 \cdot 3$

Step 3.1.8

Multiply $- 16$ by $3$ .

$16 - 16 i \sqrt{3} - 16 i \sqrt{3} - 48$

Step 3.2

Subtract $48$ from $16$ .

$- 16 i \sqrt{3} - 16 i \sqrt{3} - 32$

Step 3.3

Subtract $16 i \sqrt{3}$ from $- 16 i \sqrt{3}$ .

$- 32 i \sqrt{3} - 32$

Step 4

Reorder $- 32 i \sqrt{3}$ and $- 32$ .

$- 32 - 32 i \sqrt{3}$