Algebra Examples

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

Step 1

Expand $(3 + i) (3 - i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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 $3$ by $3$ .

$\frac{9 + 3 (- i) + i \cdot 3 + i (- i)}{- {(2 + 2 i)}^{2}}$

Step 2.1.2

Multiply $- 1$ by $3$ .

$\frac{9 - 3 i + i \cdot 3 + i (- i)}{- {(2 + 2 i)}^{2}}$

Step 2.1.3

Move $3$ to the left of $i$ .

$\frac{9 - 3 i + 3 \cdot i + i (- i)}{- {(2 + 2 i)}^{2}}$

Step 2.1.4

Multiply $i (- i)$ .

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Step 2.1.4.1

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

$\frac{9 - 3 i + 3 i - (i^{1} i)}{- {(2 + 2 i)}^{2}}$

Step 2.1.4.2

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

$\frac{9 - 3 i + 3 i - (i^{1} i^{1})}{- {(2 + 2 i)}^{2}}$

Step 2.1.4.3

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

$\frac{9 - 3 i + 3 i - i^{1 + 1}}{- {(2 + 2 i)}^{2}}$

Step 2.1.4.4

Add $1$ and $1$ .

$\frac{9 - 3 i + 3 i - i^{2}}{- {(2 + 2 i)}^{2}}$

Step 2.1.5

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

$\frac{9 - 3 i + 3 i - - 1}{- {(2 + 2 i)}^{2}}$

Step 2.1.6

Multiply $- 1$ by $- 1$ .

$\frac{9 - 3 i + 3 i + 1}{- {(2 + 2 i)}^{2}}$

Step 2.2

Add $9$ and $1$ .

$\frac{10 - 3 i + 3 i}{- {(2 + 2 i)}^{2}}$

Step 2.3

Add $- 3 i$ and $3 i$ .

$\frac{10 + 0}{- {(2 + 2 i)}^{2}}$

Step 2.4

Add $10$ and $0$ .

$\frac{10}{- {(2 + 2 i)}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite ${(2 + 2 i)}^{2}$ as $(2 + 2 i) (2 + 2 i)$ .

$\frac{10}{- ((2 + 2 i) (2 + 2 i))}$

Step 3.2

Expand $(2 + 2 i) (2 + 2 i)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{10}{- (2 (2 + 2 i) + 2 i (2 + 2 i))}$

Step 3.2.2

Apply the distributive property.

$\frac{10}{- (2 \cdot 2 + 2 (2 i) + 2 i (2 + 2 i))}$

Step 3.2.3

Apply the distributive property.

$\frac{10}{- (2 \cdot 2 + 2 (2 i) + 2 i \cdot 2 + 2 i (2 i))}$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $2$ by $2$ .

$\frac{10}{- (4 + 2 (2 i) + 2 i \cdot 2 + 2 i (2 i))}$

Step 3.3.1.2

Multiply $2$ by $2$ .

$\frac{10}{- (4 + 4 i + 2 i \cdot 2 + 2 i (2 i))}$

Step 3.3.1.3

Multiply $2$ by $2$ .

$\frac{10}{- (4 + 4 i + 4 i + 2 i (2 i))}$

Step 3.3.1.4

Multiply $2 i (2 i)$ .

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Step 3.3.1.4.1

Multiply $2$ by $2$ .

$\frac{10}{- (4 + 4 i + 4 i + 4 i i)}$

Step 3.3.1.4.2

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

$\frac{10}{- (4 + 4 i + 4 i + 4 (i^{1} i))}$

Step 3.3.1.4.3

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

$\frac{10}{- (4 + 4 i + 4 i + 4 (i^{1} i^{1}))}$

Step 3.3.1.4.4

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

$\frac{10}{- (4 + 4 i + 4 i + 4 i^{1 + 1})}$

Step 3.3.1.4.5

Add $1$ and $1$ .

$\frac{10}{- (4 + 4 i + 4 i + 4 i^{2})}$

Step 3.3.1.5

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

$\frac{10}{- (4 + 4 i + 4 i + 4 \cdot - 1)}$

Step 3.3.1.6

Multiply $4$ by $- 1$ .

$\frac{10}{- (4 + 4 i + 4 i - 4)}$

Step 3.3.2

Subtract $4$ from $4$ .

$\frac{10}{- (0 + 4 i + 4 i)}$

Step 3.3.3

Add $0$ and $4 i$ .

$\frac{10}{- (4 i + 4 i)}$

Step 3.3.4

Add $4 i$ and $4 i$ .

$\frac{10}{- (8 i)}$

Step 3.4

Multiply $8$ by $- 1$ .

$\frac{10}{- 8 i}$

Step 4

Cancel the common factor of $10$ and $- 8$ .

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Step 4.1

Factor $2$ out of $10$ .

$\frac{2 (5)}{- 8 i}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $- 8 i$ .

$\frac{2 (5)}{2 (- 4 i)}$

Step 4.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 (- 4 i)}$

Step 4.2.3

Rewrite the expression.

$\frac{5}{- 4 i}$

Step 5

Multiply the numerator and denominator of $\frac{5}{- 4 i}$ by the conjugate of $- 4 i$ to make the denominator real.

$\frac{5}{- 4 i} \cdot \frac{i}{i}$

Step 6

Multiply.

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Step 6.1

Combine.

$\frac{5 i}{- 4 i i}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Add parentheses.

$\frac{5 i}{- 4 (i i)}$

Step 6.2.2

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

$\frac{5 i}{- 4 (i^{1} i)}$

Step 6.2.3

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

$\frac{5 i}{- 4 (i^{1} i^{1})}$

Step 6.2.4

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

$\frac{5 i}{- 4 i^{1 + 1}}$

Step 6.2.5

Add $1$ and $1$ .

$\frac{5 i}{- 4 i^{2}}$

Step 6.2.6

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

$\frac{5 i}{- 4 \cdot - 1}$

Step 7

Multiply $- 4$ by $- 1$ .

$\frac{5 i}{4}$