Basic Math Examples

\frac{1 + 9 i}{5 + 10 i}

$\frac{1 + 9 i}{5 + 10 i}$

Step 1

Multiply the numerator and denominator of

$\frac{1 + 9 i}{5 + 10 i}$ by the conjugate of

$5 + 10 i$ to make the denominator real.

$\frac{1 + 9 i}{5 + 10 i} \cdot \frac{5 - 10 i}{5 - 10 i}$

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

$\frac{(1 + 9 i) (5 - 10 i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Expand

$(1 + 9 i) (5 - 10 i)$ using the FOIL Method.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$\frac{1 (5 - 10 i) + 9 i (5 - 10 i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{1 \cdot 5 + 1 (- 10 i) + 9 i (5 - 10 i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{1 \cdot 5 + 1 (- 10 i) + 9 i \cdot 5 + 9 i (- 10 i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2

Simplify and combine like terms.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Multiply

$5$ by

$1$ .

$\frac{5 + 1 (- 10 i) + 9 i \cdot 5 + 9 i (- 10 i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.2

Multiply

$- 10 i$ by

$1$ .

$\frac{5 - 10 i + 9 i \cdot 5 + 9 i (- 10 i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.3

Multiply

$5$ by

$9$ .

$\frac{5 - 10 i + 45 i + 9 i (- 10 i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.4

Multiply

$9 i (- 10 i)$ .

Tap for more steps...

Step 2.2.2.1.4.1

Multiply

$- 10$ by

$9$ .

$\frac{5 - 10 i + 45 i - 90 i i}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.4.2

Raise

$i$ to the power of

$1$ .

$\frac{5 - 10 i + 45 i - 90 (i^{1} i)}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.4.3

Raise

$i$ to the power of

$1$ .

$\frac{5 - 10 i + 45 i - 90 (i^{1} i^{1})}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 - 10 i + 45 i - 90 i^{1 + 1}}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.4.5

Add

$1$ and

$1$ .

$\frac{5 - 10 i + 45 i - 90 i^{2}}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{5 - 10 i + 45 i - 90 \cdot - 1}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.1.6

Multiply

$- 90$ by

$- 1$ .

$\frac{5 - 10 i + 45 i + 90}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.2

Add

$5$ and

$90$ .

$\frac{95 - 10 i + 45 i}{(5 + 10 i) (5 - 10 i)}$

Step 2.2.2.3

Add

$- 10 i$ and

$45 i$ .

$\frac{95 + 35 i}{(5 + 10 i) (5 - 10 i)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Expand

$(5 + 10 i) (5 - 10 i)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

$\frac{95 + 35 i}{5 (5 - 10 i) + 10 i (5 - 10 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{95 + 35 i}{5 \cdot 5 + 5 (- 10 i) + 10 i (5 - 10 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{95 + 35 i}{5 \cdot 5 + 5 (- 10 i) + 10 i \cdot 5 + 10 i (- 10 i)}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply

$5$ by

$5$ .

$\frac{95 + 35 i}{25 + 5 (- 10 i) + 10 i \cdot 5 + 10 i (- 10 i)}$

Step 2.3.2.2

Multiply

$- 10$ by

$5$ .

$\frac{95 + 35 i}{25 - 50 i + 10 i \cdot 5 + 10 i (- 10 i)}$

Step 2.3.2.3

Multiply

$5$ by

$10$ .

$\frac{95 + 35 i}{25 - 50 i + 50 i + 10 i (- 10 i)}$

Step 2.3.2.4

Multiply

$- 10$ by

$10$ .

$\frac{95 + 35 i}{25 - 50 i + 50 i - 100 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{95 + 35 i}{25 - 50 i + 50 i - 100 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{95 + 35 i}{25 - 50 i + 50 i - 100 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{95 + 35 i}{25 - 50 i + 50 i - 100 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{95 + 35 i}{25 - 50 i + 50 i - 100 i^{2}}$

Step 2.3.2.9

Add

$- 50 i$ and

$50 i$ .

$\frac{95 + 35 i}{25 + 0 - 100 i^{2}}$

Step 2.3.2.10

Add

$25$ and

$0$ .

$\frac{95 + 35 i}{25 - 100 i^{2}}$

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{95 + 35 i}{25 - 100 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 100$ by

$- 1$ .

$\frac{95 + 35 i}{25 + 100}$

Step 2.3.4

Add

$25$ and

$100$ .

$\frac{95 + 35 i}{125}$

Step 3

Cancel the common factor of

$95 + 35 i$ and

$125$ .

Tap for more steps...

Step 3.1

Factor

$5$ out of

$95$ .

$\frac{5 (19) + 35 i}{125}$

Step 3.2

Factor

$5$ out of

$35 i$ .

$\frac{5 (19) + 5 (7 i)}{125}$

Step 3.3

Factor

$5$ out of

$5 (19) + 5 (7 i)$ .

$\frac{5 (19 + 7 i)}{125}$

Step 3.4

Cancel the common factors.

Tap for more steps...

Step 3.4.1

Factor

$5$ out of

$125$ .

$\frac{5 (19 + 7 i)}{5 \cdot 25}$

Step 3.4.2

Cancel the common factor.

$\frac{5 (19 + 7 i)}{5 \cdot 25}$

Step 3.4.3

Rewrite the expression.

$\frac{19 + 7 i}{25}$

Step 4

Split the fraction

$\frac{19 + 7 i}{25}$ into two fractions.

$\frac{19}{25} + \frac{7 i}{25}$