Finite Math Examples

$\frac{{(\frac{5 - i}{7 + i})}^{2}}{2 \cdot \frac{i}{3 + i}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $\frac{5 - i}{7 + i}$ .

$\frac{\frac{{(5 - i)}^{2}}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.2

Rewrite ${(5 - i)}^{2}$ as $(5 - i) (5 - i)$ .

$\frac{\frac{(5 - i) (5 - i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.3

Expand $(5 - i) (5 - i)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$\frac{\frac{5 (5 - i) - i (5 - i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.3.2

Apply the distributive property.

$\frac{\frac{5 \cdot 5 + 5 (- i) - i (5 - i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.3.3

Apply the distributive property.

$\frac{\frac{5 \cdot 5 + 5 (- i) - i \cdot 5 - i (- i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $5$ by $5$ .

$\frac{\frac{25 + 5 (- i) - i \cdot 5 - i (- i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.2

Multiply $- 1$ by $5$ .

$\frac{\frac{25 - 5 i - i \cdot 5 - i (- i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.3

Multiply $5$ by $- 1$ .

$\frac{\frac{25 - 5 i - 5 i - i (- i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.4

Multiply $- i (- i)$ .

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Step 1.4.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{25 - 5 i - 5 i + 1 i i}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.4.2

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

$\frac{\frac{25 - 5 i - 5 i + 1 (i^{1} i)}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.4.3

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

$\frac{\frac{25 - 5 i - 5 i + 1 (i^{1} i^{1})}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.4.4

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

$\frac{\frac{25 - 5 i - 5 i + 1 i^{1 + 1}}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.4.5

Add $1$ and $1$ .

$\frac{\frac{25 - 5 i - 5 i + 1 i^{2}}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.4.6

Multiply $i^{2}$ by $1$ .

$\frac{\frac{25 - 5 i - 5 i + i^{2}}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.1.5

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

$\frac{\frac{25 - 5 i - 5 i - 1}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.2

Subtract $1$ from $25$ .

$\frac{\frac{24 - 5 i - 5 i}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.4.3

Subtract $5 i$ from $- 5 i$ .

$\frac{\frac{24 - 10 i}{{(7 + i)}^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.5

Rewrite ${(7 + i)}^{2}$ as $(7 + i) (7 + i)$ .

$\frac{\frac{24 - 10 i}{(7 + i) (7 + i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.6

Expand $(7 + i) (7 + i)$ using the FOIL Method.

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Step 1.6.1

Apply the distributive property.

$\frac{\frac{24 - 10 i}{7 (7 + i) + i (7 + i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.6.2

Apply the distributive property.

$\frac{\frac{24 - 10 i}{7 \cdot 7 + 7 i + i (7 + i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.6.3

Apply the distributive property.

$\frac{\frac{24 - 10 i}{7 \cdot 7 + 7 i + i \cdot 7 + i i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $7$ by $7$ .

$\frac{\frac{24 - 10 i}{49 + 7 i + i \cdot 7 + i i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.1.2

Move $7$ to the left of $i$ .

$\frac{\frac{24 - 10 i}{49 + 7 i + 7 \cdot i + i i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.1.3

Multiply $i i$ .

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Step 1.7.1.3.1

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

$\frac{\frac{24 - 10 i}{49 + 7 i + 7 i + i^{1} i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.1.3.2

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

$\frac{\frac{24 - 10 i}{49 + 7 i + 7 i + i^{1} i^{1}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.1.3.3

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

$\frac{\frac{24 - 10 i}{49 + 7 i + 7 i + i^{1 + 1}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.1.3.4

Add $1$ and $1$ .

$\frac{\frac{24 - 10 i}{49 + 7 i + 7 i + i^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.1.4

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

$\frac{\frac{24 - 10 i}{49 + 7 i + 7 i - 1}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.2

Subtract $1$ from $49$ .

$\frac{\frac{24 - 10 i}{48 + 7 i + 7 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.7.3

Add $7 i$ and $7 i$ .

$\frac{\frac{24 - 10 i}{48 + 14 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8

Cancel the common factor of $24 - 10 i$ and $48 + 14 i$ .

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Step 1.8.1

Factor $2$ out of $24$ .

$\frac{\frac{2 (12) - 10 i}{48 + 14 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8.2

Factor $2$ out of $- 10 i$ .

$\frac{\frac{2 (12) + 2 (- 5 i)}{48 + 14 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8.3

Factor $2$ out of $2 (12) + 2 (- 5 i)$ .

$\frac{\frac{2 (12 - 5 i)}{48 + 14 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8.4

Cancel the common factors.

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Step 1.8.4.1

Factor $2$ out of $48$ .

$\frac{\frac{2 (12 - 5 i)}{2 \cdot 24 + 14 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8.4.2

Factor $2$ out of $14 i$ .

$\frac{\frac{2 (12 - 5 i)}{2 \cdot 24 + 2 (7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8.4.3

Factor $2$ out of $2 (24) + 2 (7 i)$ .

$\frac{\frac{2 (12 - 5 i)}{2 (24 + 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8.4.4

Cancel the common factor.

$\frac{\frac{2 (12 - 5 i)}{2 (24 + 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.8.4.5

Rewrite the expression.

$\frac{\frac{12 - 5 i}{24 + 7 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.9

Multiply the numerator and denominator of $\frac{12 - 5 i}{24 + 7 i}$ by the conjugate of $24 + 7 i$ to make the denominator real.

$\frac{\frac{12 - 5 i}{24 + 7 i} \cdot \frac{24 - 7 i}{24 - 7 i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10

Multiply.

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Step 1.10.1

Combine.

$\frac{\frac{(12 - 5 i) (24 - 7 i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2

Simplify the numerator.

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Step 1.10.2.1

Expand $(12 - 5 i) (24 - 7 i)$ using the FOIL Method.

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Step 1.10.2.1.1

Apply the distributive property.

$\frac{\frac{12 (24 - 7 i) - 5 i (24 - 7 i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.1.2

Apply the distributive property.

$\frac{\frac{12 \cdot 24 + 12 (- 7 i) - 5 i (24 - 7 i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.1.3

Apply the distributive property.

$\frac{\frac{12 \cdot 24 + 12 (- 7 i) - 5 i \cdot 24 - 5 i (- 7 i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2

Simplify and combine like terms.

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Step 1.10.2.2.1

Simplify each term.

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Step 1.10.2.2.1.1

Multiply $12$ by $24$ .

$\frac{\frac{288 + 12 (- 7 i) - 5 i \cdot 24 - 5 i (- 7 i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.2

Multiply $- 7$ by $12$ .

$\frac{\frac{288 - 84 i - 5 i \cdot 24 - 5 i (- 7 i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.3

Multiply $24$ by $- 5$ .

$\frac{\frac{288 - 84 i - 120 i - 5 i (- 7 i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.4

Multiply $- 5 i (- 7 i)$ .

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Step 1.10.2.2.1.4.1

Multiply $- 7$ by $- 5$ .

$\frac{\frac{288 - 84 i - 120 i + 35 i i}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.4.2

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

$\frac{\frac{288 - 84 i - 120 i + 35 (i^{1} i)}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.4.3

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

$\frac{\frac{288 - 84 i - 120 i + 35 (i^{1} i^{1})}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.4.4

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

$\frac{\frac{288 - 84 i - 120 i + 35 i^{1 + 1}}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.4.5

Add $1$ and $1$ .

$\frac{\frac{288 - 84 i - 120 i + 35 i^{2}}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.5

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

$\frac{\frac{288 - 84 i - 120 i + 35 \cdot - 1}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.1.6

Multiply $35$ by $- 1$ .

$\frac{\frac{288 - 84 i - 120 i - 35}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.2

Subtract $35$ from $288$ .

$\frac{\frac{253 - 84 i - 120 i}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.2.2.3

Subtract $120 i$ from $- 84 i$ .

$\frac{\frac{253 - 204 i}{(24 + 7 i) (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3

Simplify the denominator.

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Step 1.10.3.1

Expand $(24 + 7 i) (24 - 7 i)$ using the FOIL Method.

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Step 1.10.3.1.1

Apply the distributive property.

$\frac{\frac{253 - 204 i}{24 (24 - 7 i) + 7 i (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.1.2

Apply the distributive property.

$\frac{\frac{253 - 204 i}{24 \cdot 24 + 24 (- 7 i) + 7 i (24 - 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.1.3

Apply the distributive property.

$\frac{\frac{253 - 204 i}{24 \cdot 24 + 24 (- 7 i) + 7 i \cdot 24 + 7 i (- 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2

Simplify.

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Step 1.10.3.2.1

Multiply $24$ by $24$ .

$\frac{\frac{253 - 204 i}{576 + 24 (- 7 i) + 7 i \cdot 24 + 7 i (- 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.2

Multiply $- 7$ by $24$ .

$\frac{\frac{253 - 204 i}{576 - 168 i + 7 i \cdot 24 + 7 i (- 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.3

Multiply $24$ by $7$ .

$\frac{\frac{253 - 204 i}{576 - 168 i + 168 i + 7 i (- 7 i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.4

Multiply $- 7$ by $7$ .

$\frac{\frac{253 - 204 i}{576 - 168 i + 168 i - 49 i i}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.5

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

$\frac{\frac{253 - 204 i}{576 - 168 i + 168 i - 49 (i^{1} i)}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.6

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

$\frac{\frac{253 - 204 i}{576 - 168 i + 168 i - 49 (i^{1} i^{1})}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.7

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

$\frac{\frac{253 - 204 i}{576 - 168 i + 168 i - 49 i^{1 + 1}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.8

Add $1$ and $1$ .

$\frac{\frac{253 - 204 i}{576 - 168 i + 168 i - 49 i^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.9

Add $- 168 i$ and $168 i$ .

$\frac{\frac{253 - 204 i}{576 + 0 - 49 i^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.2.10

Add $576$ and $0$ .

$\frac{\frac{253 - 204 i}{576 - 49 i^{2}}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.3

Simplify each term.

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Step 1.10.3.3.1

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

$\frac{\frac{253 - 204 i}{576 - 49 \cdot - 1}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.3.2

Multiply $- 49$ by $- 1$ .

$\frac{\frac{253 - 204 i}{576 + 49}}{2 \cdot \frac{i}{3 + i}}$

Step 1.10.3.4

Add $576$ and $49$ .

$\frac{\frac{253 - 204 i}{625}}{2 \cdot \frac{i}{3 + i}}$

$\frac{\frac{253 - 204 i}{625}}{\frac{2 i}{3 + i}}$

Step 2

Simplify the denominator.

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Step 2.1

Multiply the numerator and denominator of $\frac{2 i}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$\frac{\frac{253 - 204 i}{625}}{\frac{2 i}{3 + 1 i} \cdot \frac{3 - i}{3 - i}}$

Step 2.2

Multiply.

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Step 2.2.1

Combine.

$\frac{\frac{253 - 204 i}{625}}{\frac{2 i (3 - i)}{(3 + 1 i) (3 - i)}}$

Step 2.2.2

Simplify the numerator.

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Step 2.2.2.1

Apply the distributive property.

$\frac{\frac{253 - 204 i}{625}}{\frac{2 i \cdot 3 + 2 i (- i)}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.2

Multiply $3$ by $2$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i + 2 i (- i)}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.3

Multiply $2 i (- i)$ .

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Step 2.2.2.3.1

Multiply $- 1$ by $2$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i - 2 i i}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.3.2

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

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i - 2 (i^{1} i)}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.3.3

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

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i - 2 (i^{1} i^{1})}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.3.4

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

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i - 2 i^{1 + 1}}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.3.5

Add $1$ and $1$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i - 2 i^{2}}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.4

Simplify each term.

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Step 2.2.2.4.1

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

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i - 2 \cdot - 1}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.4.2

Multiply $- 2$ by $- 1$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{6 i + 2}{(3 + 1 i) (3 - i)}}$

Step 2.2.2.5

Reorder $6 i$ and $2$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{(3 + 1 i) (3 - i)}}$

Step 2.2.3

Simplify the denominator.

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Step 2.2.3.1

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

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Step 2.2.3.1.1

Apply the distributive property.

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{3 (3 - i) + 1 i (3 - i)}}$

Step 2.2.3.1.2

Apply the distributive property.

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}}$

Step 2.2.3.1.3

Apply the distributive property.

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}}$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Multiply $3$ by $3$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}}$

Step 2.2.3.2.2

Multiply $- 1$ by $3$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}}$

Step 2.2.3.2.3

Multiply $3$ by $1$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - 3 i + 3 i + 1 i (- i)}}$

Step 2.2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - 3 i + 3 i - i i}}$

Step 2.2.3.2.5

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

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - 3 i + 3 i - (i^{1} i)}}$

Step 2.2.3.2.6

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

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - 3 i + 3 i - (i^{1} i^{1})}}$

Step 2.2.3.2.7

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

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - 3 i + 3 i - i^{1 + 1}}}$

Step 2.2.3.2.8

Add $1$ and $1$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - 3 i + 3 i - i^{2}}}$

Step 2.2.3.2.9

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

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 + 0 - i^{2}}}$

Step 2.2.3.2.10

Add $9$ and $0$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - i^{2}}}$

Step 2.2.3.3

Simplify each term.

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Step 2.2.3.3.1

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

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 - - 1}}$

Step 2.2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{9 + 1}}$

Step 2.2.3.4

Add $9$ and $1$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 + 6 i}{10}}$

Step 2.3

Cancel the common factor of $2 + 6 i$ and $10$ .

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Step 2.3.1

Factor $2$ out of $2$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 (1) + 6 i}{10}}$

Step 2.3.2

Factor $2$ out of $6 i$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 (1) + 2 (3 i)}{10}}$

Step 2.3.3

Factor $2$ out of $2 (1) + 2 (3 i)$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 (1 + 3 i)}{10}}$

Step 2.3.4

Cancel the common factors.

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Step 2.3.4.1

Factor $2$ out of $10$ .

$\frac{\frac{253 - 204 i}{625}}{\frac{2 (1 + 3 i)}{2 \cdot 5}}$

Step 2.3.4.2

Cancel the common factor.

$\frac{\frac{253 - 204 i}{625}}{\frac{2 (1 + 3 i)}{2 \cdot 5}}$

Step 2.3.4.3

Rewrite the expression.

$\frac{\frac{253 - 204 i}{625}}{\frac{1 + 3 i}{5}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{253 - 204 i}{625} \cdot \frac{5}{1 + 3 i}$

Step 4

Cancel the common factor of $5$ .

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Step 4.1

Factor $5$ out of $625$ .

$\frac{253 - 204 i}{5 (125)} \cdot \frac{5}{1 + 3 i}$

Step 4.2

Cancel the common factor.

$\frac{253 - 204 i}{5 \cdot 125} \cdot \frac{5}{1 + 3 i}$

Step 4.3

Rewrite the expression.

$\frac{253 - 204 i}{125} \cdot \frac{1}{1 + 3 i}$

Step 5

Multiply $\frac{253 - 204 i}{125}$ by $\frac{1}{1 + 3 i}$ .

$\frac{253 - 204 i}{125 (1 + 3 i)}$

Step 6

Multiply the numerator and denominator of $\frac{253 - 204 i}{125 + 375 i}$ by the conjugate of $125 + 375 i$ to make the denominator real.

$\frac{253 - 204 i}{125 + 375 i} \cdot \frac{125 - 375 i}{125 - 375 i}$

Step 7

Multiply.

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Step 7.1

Combine.

$\frac{(253 - 204 i) (125 - 375 i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Expand $(253 - 204 i) (125 - 375 i)$ using the FOIL Method.

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Step 7.2.1.1

Apply the distributive property.

$\frac{253 (125 - 375 i) - 204 i (125 - 375 i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.1.2

Apply the distributive property.

$\frac{253 \cdot 125 + 253 (- 375 i) - 204 i (125 - 375 i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.1.3

Apply the distributive property.

$\frac{253 \cdot 125 + 253 (- 375 i) - 204 i \cdot 125 - 204 i (- 375 i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2

Simplify and combine like terms.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Multiply $253$ by $125$ .

$\frac{31625 + 253 (- 375 i) - 204 i \cdot 125 - 204 i (- 375 i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.2

Multiply $- 375$ by $253$ .

$\frac{31625 - 94875 i - 204 i \cdot 125 - 204 i (- 375 i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.3

Multiply $125$ by $- 204$ .

$\frac{31625 - 94875 i - 25500 i - 204 i (- 375 i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.4

Multiply $- 204 i (- 375 i)$ .

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Step 7.2.2.1.4.1

Multiply $- 375$ by $- 204$ .

$\frac{31625 - 94875 i - 25500 i + 76500 i i}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.4.2

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

$\frac{31625 - 94875 i - 25500 i + 76500 (i^{1} i)}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.4.3

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

$\frac{31625 - 94875 i - 25500 i + 76500 (i^{1} i^{1})}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.4.4

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

$\frac{31625 - 94875 i - 25500 i + 76500 i^{1 + 1}}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.4.5

Add $1$ and $1$ .

$\frac{31625 - 94875 i - 25500 i + 76500 i^{2}}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.5

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

$\frac{31625 - 94875 i - 25500 i + 76500 \cdot - 1}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.1.6

Multiply $76500$ by $- 1$ .

$\frac{31625 - 94875 i - 25500 i - 76500}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.2

Subtract $76500$ from $31625$ .

$\frac{- 44875 - 94875 i - 25500 i}{(125 + 375 i) (125 - 375 i)}$

Step 7.2.2.3

Subtract $25500 i$ from $- 94875 i$ .

$\frac{- 44875 - 120375 i}{(125 + 375 i) (125 - 375 i)}$

Step 7.3

Simplify the denominator.

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Step 7.3.1

Expand $(125 + 375 i) (125 - 375 i)$ using the FOIL Method.

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Step 7.3.1.1

Apply the distributive property.

$\frac{- 44875 - 120375 i}{125 (125 - 375 i) + 375 i (125 - 375 i)}$

Step 7.3.1.2

Apply the distributive property.

$\frac{- 44875 - 120375 i}{125 \cdot 125 + 125 (- 375 i) + 375 i (125 - 375 i)}$

Step 7.3.1.3

Apply the distributive property.

$\frac{- 44875 - 120375 i}{125 \cdot 125 + 125 (- 375 i) + 375 i \cdot 125 + 375 i (- 375 i)}$

Step 7.3.2

Simplify.

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Step 7.3.2.1

Multiply $125$ by $125$ .

$\frac{- 44875 - 120375 i}{15625 + 125 (- 375 i) + 375 i \cdot 125 + 375 i (- 375 i)}$

Step 7.3.2.2

Multiply $- 375$ by $125$ .

$\frac{- 44875 - 120375 i}{15625 - 46875 i + 375 i \cdot 125 + 375 i (- 375 i)}$

Step 7.3.2.3

Multiply $125$ by $375$ .

$\frac{- 44875 - 120375 i}{15625 - 46875 i + 46875 i + 375 i (- 375 i)}$

Step 7.3.2.4

Multiply $- 375$ by $375$ .

$\frac{- 44875 - 120375 i}{15625 - 46875 i + 46875 i - 140625 i i}$

Step 7.3.2.5

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

$\frac{- 44875 - 120375 i}{15625 - 46875 i + 46875 i - 140625 (i^{1} i)}$

Step 7.3.2.6

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

$\frac{- 44875 - 120375 i}{15625 - 46875 i + 46875 i - 140625 (i^{1} i^{1})}$

Step 7.3.2.7

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

$\frac{- 44875 - 120375 i}{15625 - 46875 i + 46875 i - 140625 i^{1 + 1}}$

Step 7.3.2.8

Add $1$ and $1$ .

$\frac{- 44875 - 120375 i}{15625 - 46875 i + 46875 i - 140625 i^{2}}$

Step 7.3.2.9

Add $- 46875 i$ and $46875 i$ .

$\frac{- 44875 - 120375 i}{15625 + 0 - 140625 i^{2}}$

Step 7.3.2.10

Add $15625$ and $0$ .

$\frac{- 44875 - 120375 i}{15625 - 140625 i^{2}}$

Step 7.3.3

Simplify each term.

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Step 7.3.3.1

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

$\frac{- 44875 - 120375 i}{15625 - 140625 \cdot - 1}$

Step 7.3.3.2

Multiply $- 140625$ by $- 1$ .

$\frac{- 44875 - 120375 i}{15625 + 140625}$

Step 7.3.4

Add $15625$ and $140625$ .

$\frac{- 44875 - 120375 i}{156250}$

Step 8

Cancel the common factor of $- 44875 - 120375 i$ and $156250$ .

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Step 8.1

Factor $125$ out of $- 44875$ .

$\frac{125 (- 359) - 120375 i}{156250}$

Step 8.2

Factor $125$ out of $- 120375 i$ .

$\frac{125 (- 359) + 125 (- 963 i)}{156250}$

Step 8.3

Factor $125$ out of $125 (- 359) + 125 (- 963 i)$ .

$\frac{125 (- 359 - 963 i)}{156250}$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $125$ out of $156250$ .

$\frac{125 (- 359 - 963 i)}{125 \cdot 1250}$

Step 8.4.2

Cancel the common factor.

$\frac{125 (- 359 - 963 i)}{125 \cdot 1250}$

Step 8.4.3

Rewrite the expression.

$\frac{- 359 - 963 i}{1250}$

Step 9

Split the fraction $\frac{- 359 - 963 i}{1250}$ into two fractions.

$\frac{- 359}{1250} + \frac{- 963 i}{1250}$

Step 10

Simplify each term.

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Step 10.1

Move the negative in front of the fraction.

$- \frac{359}{1250} + \frac{- 963 i}{1250}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{359}{1250} - \frac{963 i}{1250}$