Basic Math Examples

$\frac{3 + 5 i}{6 - \sqrt{2} i} \cdot \frac{6 + \sqrt{2} i}{6 + \sqrt{2} i}$

Step 1

Cancel the common factor of $6 + \sqrt{2} i$ .

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Step 1.1

Cancel the common factor.

$\frac{3 + 5 i}{6 - \sqrt{2} i} \cdot \frac{6 + \sqrt{2} i}{6 + \sqrt{2} i}$

Step 1.2

Rewrite the expression.

$\frac{3 + 5 i}{6 - \sqrt{2} i} \cdot 1$

Step 2

Multiply $\frac{3 + 5 i}{6 - \sqrt{2} i}$ by $1$ .

$\frac{3 + 5 i}{6 - \sqrt{2} i}$

Step 3

Multiply the numerator and denominator of $\frac{3 + 5 i}{6 - 1.41421356 i}$ by the conjugate of $6 - 1.41421356 i$ to make the denominator real.

$\frac{3 + 5 i}{6 - 1.41421356 i} \cdot \frac{6 + 1.41421356 i}{6 + 1.41421356 i}$

Step 4

Multiply.

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Step 4.1

Combine.

$\frac{(3 + 5 i) (6 + 1.41421356 i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Expand $(3 + 5 i) (6 + 1.41421356 i)$ using the FOIL Method.

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Step 4.2.1.1

Apply the distributive property.

$\frac{3 (6 + 1.41421356 i) + 5 i (6 + 1.41421356 i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.1.2

Apply the distributive property.

$\frac{3 \cdot 6 + 3 (1.41421356 i) + 5 i (6 + 1.41421356 i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.1.3

Apply the distributive property.

$\frac{3 \cdot 6 + 3 (1.41421356 i) + 5 i \cdot 6 + 5 i (1.41421356 i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2

Simplify and combine like terms.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $3$ by $6$ .

$\frac{18 + 3 (1.41421356 i) + 5 i \cdot 6 + 5 i (1.41421356 i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.2

Multiply $1.41421356$ by $3$ .

$\frac{18 + 4.24264068 i + 5 i \cdot 6 + 5 i (1.41421356 i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.3

Multiply $6$ by $5$ .

$\frac{18 + 4.24264068 i + 30 i + 5 i (1.41421356 i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.4

Multiply $5 i (1.41421356 i)$ .

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Step 4.2.2.1.4.1

Multiply $1.41421356$ by $5$ .

$\frac{18 + 4.24264068 i + 30 i + 7.07106781 i i}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.4.2

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

$\frac{18 + 4.24264068 i + 30 i + 7.07106781 (i^{1} i)}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.4.3

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

$\frac{18 + 4.24264068 i + 30 i + 7.07106781 (i^{1} i^{1})}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.4.4

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

$\frac{18 + 4.24264068 i + 30 i + 7.07106781 i^{1 + 1}}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.4.5

Add $1$ and $1$ .

$\frac{18 + 4.24264068 i + 30 i + 7.07106781 i^{2}}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.5

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

$\frac{18 + 4.24264068 i + 30 i + 7.07106781 \cdot - 1}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.1.6

Multiply $7.07106781$ by $- 1$ .

$\frac{18 + 4.24264068 i + 30 i - 7.07106781}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.2

Subtract $7.07106781$ from $18$ .

$\frac{10.92893218 + 4.24264068 i + 30 i}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.2.2.3

Add $4.24264068 i$ and $30 i$ .

$\frac{10.92893218 + 34.24264068 i}{(6 - 1.41421356 i) (6 + 1.41421356 i)}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Expand $(6 - 1.41421356 i) (6 + 1.41421356 i)$ using the FOIL Method.

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Step 4.3.1.1

Apply the distributive property.

$\frac{10.92893218 + 34.24264068 i}{6 (6 + 1.41421356 i) - 1.41421356 i (6 + 1.41421356 i)}$

Step 4.3.1.2

Apply the distributive property.

$\frac{10.92893218 + 34.24264068 i}{6 \cdot 6 + 6 (1.41421356 i) - 1.41421356 i (6 + 1.41421356 i)}$

Step 4.3.1.3

Apply the distributive property.

$\frac{10.92893218 + 34.24264068 i}{6 \cdot 6 + 6 (1.41421356 i) - 1.41421356 i \cdot 6 - 1.41421356 i (1.41421356 i)}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Multiply $6$ by $6$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 6 (1.41421356 i) - 1.41421356 i \cdot 6 - 1.41421356 i (1.41421356 i)}$

Step 4.3.2.2

Multiply $1.41421356$ by $6$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 8.48528137 i - 1.41421356 i \cdot 6 - 1.41421356 i (1.41421356 i)}$

Step 4.3.2.3

Multiply $6$ by $- 1.41421356$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 8.48528137 i - 8.48528137 i - 1.41421356 i (1.41421356 i)}$

Step 4.3.2.4

Multiply $1.41421356$ by $- 1.41421356$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 8.48528137 i - 8.48528137 i - 2 i i}$

Step 4.3.2.5

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

$\frac{10.92893218 + 34.24264068 i}{36 + 8.48528137 i - 8.48528137 i - 2 (i^{1} i)}$

Step 4.3.2.6

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

$\frac{10.92893218 + 34.24264068 i}{36 + 8.48528137 i - 8.48528137 i - 2 (i^{1} i^{1})}$

Step 4.3.2.7

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

$\frac{10.92893218 + 34.24264068 i}{36 + 8.48528137 i - 8.48528137 i - 2 i^{1 + 1}}$

Step 4.3.2.8

Add $1$ and $1$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 8.48528137 i - 8.48528137 i - 2 i^{2}}$

Step 4.3.2.9

Subtract $8.48528137 i$ from $8.48528137 i$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 0 i - 2 i^{2}}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Multiply $0$ by $i$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 0 - 2 i^{2}}$

Step 4.3.3.2

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

$\frac{10.92893218 + 34.24264068 i}{36 + 0 - 2 \cdot - 1}$

Step 4.3.3.3

Multiply $- 2$ by $- 1$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 0 + 2}$

Step 4.3.4

Add $36$ and $0$ .

$\frac{10.92893218 + 34.24264068 i}{36 + 2}$

Step 4.3.5

Add $36$ and $2$ .

$\frac{10.92893218 + 34.24264068 i}{38}$

Step 5

Rewrite $10.92893218$ as $1 (10.92893218)$ .

$\frac{1 (10.92893218) + 34.24264068 i}{38}$

Step 6

Factor $1$ out of $34.24264068 i$ .

$\frac{1 (10.92893218) + 1 (34.24264068 i)}{38}$

Step 7

Factor $1$ out of $1 (10.92893218) + 1 (34.24264068 i)$ .

$\frac{1 (10.92893218 + 34.24264068 i)}{38}$

Step 8

Factor $38$ out of $38$ .

$\frac{1 (10.92893218 + 34.24264068 i)}{38 (1)}$

Step 9

Separate fractions.

$\frac{1}{38} \cdot \frac{10.92893218 + 34.24264068 i}{1}$

Step 10

Divide $1$ by $38$ .

$0.02631578 \frac{10.92893218 + 34.24264068 i}{1}$

Step 11

Divide $10.92893218 + 34.24264068 i$ by $1$ .

$0.02631578 (10.92893218 + 34.24264068 i)$

Step 12

Apply the distributive property.

$0.02631578 \cdot 10.92893218 + 0.02631578 (34.24264068 i)$

Step 13

Multiply $0.02631578$ by $10.92893218$ .

$0.28760347 + 0.02631578 (34.24264068 i)$

Step 14

Multiply $34.24264068$ by $0.02631578$ .

$0.28760347 + 0.90112212 i$