Basic Math Examples

$(54 + 158 i) \div (11 + 7 i)$

Step 1

Rewrite the division as a fraction.

$\frac{54 + 158 i}{11 + 7 i}$

Step 2

Multiply the numerator and denominator of $\frac{54 + 158 i}{11 + 7 i}$ by the conjugate of $11 + 7 i$ to make the denominator real.

$\frac{54 + 158 i}{11 + 7 i} \cdot \frac{11 - 7 i}{11 - 7 i}$

Step 3

Multiply.

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Step 3.1

Combine.

$\frac{(54 + 158 i) (11 - 7 i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Expand $(54 + 158 i) (11 - 7 i)$ using the FOIL Method.

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Step 3.2.1.1

Apply the distributive property.

$\frac{54 (11 - 7 i) + 158 i (11 - 7 i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.1.2

Apply the distributive property.

$\frac{54 \cdot 11 + 54 (- 7 i) + 158 i (11 - 7 i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.1.3

Apply the distributive property.

$\frac{54 \cdot 11 + 54 (- 7 i) + 158 i \cdot 11 + 158 i (- 7 i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2

Simplify and combine like terms.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $54$ by $11$ .

$\frac{594 + 54 (- 7 i) + 158 i \cdot 11 + 158 i (- 7 i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.2

Multiply $- 7$ by $54$ .

$\frac{594 - 378 i + 158 i \cdot 11 + 158 i (- 7 i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.3

Multiply $11$ by $158$ .

$\frac{594 - 378 i + 1738 i + 158 i (- 7 i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.4

Multiply $158 i (- 7 i)$ .

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Step 3.2.2.1.4.1

Multiply $- 7$ by $158$ .

$\frac{594 - 378 i + 1738 i - 1106 i i}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.4.2

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

$\frac{594 - 378 i + 1738 i - 1106 (i^{1} i)}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.4.3

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

$\frac{594 - 378 i + 1738 i - 1106 (i^{1} i^{1})}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.4.4

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

$\frac{594 - 378 i + 1738 i - 1106 i^{1 + 1}}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.4.5

Add $1$ and $1$ .

$\frac{594 - 378 i + 1738 i - 1106 i^{2}}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.5

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

$\frac{594 - 378 i + 1738 i - 1106 \cdot - 1}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.1.6

Multiply $- 1106$ by $- 1$ .

$\frac{594 - 378 i + 1738 i + 1106}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.2

Add $594$ and $1106$ .

$\frac{1700 - 378 i + 1738 i}{(11 + 7 i) (11 - 7 i)}$

Step 3.2.2.3

Add $- 378 i$ and $1738 i$ .

$\frac{1700 + 1360 i}{(11 + 7 i) (11 - 7 i)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(11 + 7 i) (11 - 7 i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$\frac{1700 + 1360 i}{11 (11 - 7 i) + 7 i (11 - 7 i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{1700 + 1360 i}{11 \cdot 11 + 11 (- 7 i) + 7 i (11 - 7 i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{1700 + 1360 i}{11 \cdot 11 + 11 (- 7 i) + 7 i \cdot 11 + 7 i (- 7 i)}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $11$ by $11$ .

$\frac{1700 + 1360 i}{121 + 11 (- 7 i) + 7 i \cdot 11 + 7 i (- 7 i)}$

Step 3.3.2.2

Multiply $- 7$ by $11$ .

$\frac{1700 + 1360 i}{121 - 77 i + 7 i \cdot 11 + 7 i (- 7 i)}$

Step 3.3.2.3

Multiply $11$ by $7$ .

$\frac{1700 + 1360 i}{121 - 77 i + 77 i + 7 i (- 7 i)}$

Step 3.3.2.4

Multiply $- 7$ by $7$ .

$\frac{1700 + 1360 i}{121 - 77 i + 77 i - 49 i i}$

Step 3.3.2.5

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

$\frac{1700 + 1360 i}{121 - 77 i + 77 i - 49 (i^{1} i)}$

Step 3.3.2.6

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

$\frac{1700 + 1360 i}{121 - 77 i + 77 i - 49 (i^{1} i^{1})}$

Step 3.3.2.7

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

$\frac{1700 + 1360 i}{121 - 77 i + 77 i - 49 i^{1 + 1}}$

Step 3.3.2.8

Add $1$ and $1$ .

$\frac{1700 + 1360 i}{121 - 77 i + 77 i - 49 i^{2}}$

Step 3.3.2.9

Add $- 77 i$ and $77 i$ .

$\frac{1700 + 1360 i}{121 + 0 - 49 i^{2}}$

Step 3.3.2.10

Add $121$ and $0$ .

$\frac{1700 + 1360 i}{121 - 49 i^{2}}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

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

$\frac{1700 + 1360 i}{121 - 49 \cdot - 1}$

Step 3.3.3.2

Multiply $- 49$ by $- 1$ .

$\frac{1700 + 1360 i}{121 + 49}$

Step 3.3.4

Add $121$ and $49$ .

$\frac{1700 + 1360 i}{170}$

Step 4

Cancel the common factor of $1700 + 1360 i$ and $170$ .

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Step 4.1

Factor $170$ out of $1700$ .

$\frac{170 \cdot 10 + 1360 i}{170}$

Step 4.2

Factor $170$ out of $1360 i$ .

$\frac{170 \cdot 10 + 170 (8 i)}{170}$

Step 4.3

Factor $170$ out of $170 (10) + 170 (8 i)$ .

$\frac{170 (10 + 8 i)}{170}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $170$ out of $170$ .

$\frac{170 (10 + 8 i)}{170 (1)}$

Step 4.4.2

Cancel the common factor.

$\frac{170 (10 + 8 i)}{170 \cdot 1}$

Step 4.4.3

Rewrite the expression.

$\frac{10 + 8 i}{1}$

Step 4.4.4

Divide $10 + 8 i$ by $1$ .

$10 + 8 i$