Algebra Examples

$\frac{4 i}{\sqrt{2} + i}$

Step 1

Multiply the numerator and denominator of $\frac{4 i}{1.41421356 + 1 i}$ by the conjugate of $1.41421356 + 1 i$ to make the denominator real.

$\frac{4 i}{1.41421356 + 1 i} \cdot \frac{1.41421356 - i}{1.41421356 - i}$

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

$\frac{4 i (1.41421356 - i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Apply the distributive property.

$\frac{4 i \cdot 1.41421356 + 4 i (- i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.2

Multiply $1.41421356$ by $4$ .

$\frac{5.65685424 i + 4 i (- i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.3

Multiply $4 i (- i)$ .

Tap for more steps...

Step 2.2.3.1

Multiply $- 1$ by $4$ .

$\frac{5.65685424 i - 4 i i}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.3.2

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

$\frac{5.65685424 i - 4 (i^{1} i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.3.3

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

$\frac{5.65685424 i - 4 (i^{1} i^{1})}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.3.4

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

$\frac{5.65685424 i - 4 i^{1 + 1}}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.3.5

Add $1$ and $1$ .

$\frac{5.65685424 i - 4 i^{2}}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.4

Simplify each term.

Tap for more steps...

Step 2.2.4.1

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

$\frac{5.65685424 i - 4 \cdot - 1}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.4.2

Multiply $- 4$ by $- 1$ .

$\frac{5.65685424 i + 4}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.2.5

Reorder $5.65685424 i$ and $4$ .

$\frac{4 + 5.65685424 i}{(1.41421356 + 1 i) (1.41421356 - i)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Expand $(1.41421356 + 1 i) (1.41421356 - i)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

$\frac{4 + 5.65685424 i}{1.41421356 (1.41421356 - i) + 1 i (1.41421356 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{4 + 5.65685424 i}{1.41421356 \cdot 1.41421356 + 1.41421356 (- i) + 1 i (1.41421356 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{4 + 5.65685424 i}{1.41421356 \cdot 1.41421356 + 1.41421356 (- i) + 1 i \cdot 1.41421356 + 1 i (- i)}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply $1.41421356$ by $1.41421356$ .

$\frac{4 + 5.65685424 i}{2 + 1.41421356 (- i) + 1 i \cdot 1.41421356 + 1 i (- i)}$

Step 2.3.2.2

Multiply $- 1$ by $1.41421356$ .

$\frac{4 + 5.65685424 i}{2 - 1.41421356 i + 1 i \cdot 1.41421356 + 1 i (- i)}$

Step 2.3.2.3

Multiply $1.41421356$ by $1$ .

$\frac{4 + 5.65685424 i}{2 - 1.41421356 i + 1.41421356 i + 1 i (- i)}$

Step 2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{4 + 5.65685424 i}{2 - 1.41421356 i + 1.41421356 i - i i}$

Step 2.3.2.5

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

$\frac{4 + 5.65685424 i}{2 - 1.41421356 i + 1.41421356 i - (i^{1} i)}$

Step 2.3.2.6

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

$\frac{4 + 5.65685424 i}{2 - 1.41421356 i + 1.41421356 i - (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{4 + 5.65685424 i}{2 - 1.41421356 i + 1.41421356 i - i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{4 + 5.65685424 i}{2 - 1.41421356 i + 1.41421356 i - i^{2}}$

Step 2.3.2.9

Add $- 1.41421356 i$ and $1.41421356 i$ .

$\frac{4 + 5.65685424 i}{2 + 0 i - i^{2}}$

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

Multiply $0$ by $i$ .

$\frac{4 + 5.65685424 i}{2 + 0 - i^{2}}$

Step 2.3.3.2

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

$\frac{4 + 5.65685424 i}{2 + 0 - - 1}$

Step 2.3.3.3

Multiply $- 1$ by $- 1$ .

$\frac{4 + 5.65685424 i}{2 + 0 + 1}$

Step 2.3.4

Add $2$ and $0$ .

$\frac{4 + 5.65685424 i}{2 + 1}$

Step 2.3.5

Add $2$ and $1$ .

$\frac{4 + 5.65685424 i}{3}$

Step 3

Rewrite $4$ as $1 (4)$ .

$\frac{1 (4) + 5.65685424 i}{3}$

Step 4

Factor $1$ out of $5.65685424 i$ .

$\frac{1 (4) + 1 (5.65685424 i)}{3}$

Step 5

Factor $1$ out of $1 (4) + 1 (5.65685424 i)$ .

$\frac{1 (4 + 5.65685424 i)}{3}$

Step 6

Factor $3$ out of $3$ .

$\frac{1 (4 + 5.65685424 i)}{3 (1)}$

Step 7

Separate fractions.

$\frac{1}{3} \cdot \frac{4 + 5.65685424 i}{1}$

Step 8

Divide $1$ by $3$ .

$0.33333333 \frac{4 + 5.65685424 i}{1}$

Step 9

Divide $4 + 5.65685424 i$ by $1$ .

$0.33333333 (4 + 5.65685424 i)$

Step 10

Apply the distributive property.

$0.33333333 \cdot 4 + 0.33333333 (5.65685424 i)$

Step 11

Multiply $0.33333333$ by $4$ .

$1.33333333 + 0.33333333 (5.65685424 i)$

Step 12

Multiply $5.65685424$ by $0.33333333$ .

$1.33333333 + 1.88561808 i$