Algebra Examples

$\frac{(1 + 2 \sqrt{- 3})}{(1 + \sqrt{- 3})}$

Step 1

Pull out imaginary unit $i$ .

Tap for more steps...

Step 1.1

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{1 + 2 \sqrt{- 1 (3)}}{1 + \sqrt{- 3}}$

Step 1.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$\frac{1 + 2 (\sqrt{- 1} \cdot \sqrt{3})}{1 + \sqrt{- 3}}$

Step 1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{1 + 2 (i \cdot \sqrt{3})}{1 + \sqrt{- 3}}$

Step 1.4

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{1 + 2 (i \cdot \sqrt{3})}{1 + \sqrt{- 1 (3)}}$

Step 1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$\frac{1 + 2 (i \cdot \sqrt{3})}{1 + \sqrt{- 1} \cdot \sqrt{3}}$

Step 1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{1 + 2 (i \cdot \sqrt{3})}{1 + i \cdot \sqrt{3}}$

Step 2

Multiply the numerator and denominator of $\frac{1 + 2 (i \cdot \sqrt{3})}{1 + 1.7320508 \cdot i}$ by the conjugate of $1 + 1.7320508 \cdot i$ to make the denominator real.

$\frac{1 + 2 (i \cdot \sqrt{3})}{1 + 1.7320508 \cdot i} \cdot \frac{1 - 1.7320508 i}{1 - 1.7320508 i}$

Step 3

Multiply.

Tap for more steps...

Step 3.1

Combine.

$\frac{(1 + 2 (i \cdot \sqrt{3})) (1 - 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2

Simplify the numerator.

Tap for more steps...

Step 3.2.1

Remove parentheses.

$\frac{(1 + 2 i \sqrt{3}) (1 - 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.2

Expand $(1 + 2 i \sqrt{3}) (1 - 1.7320508 i)$ using the FOIL Method.

Tap for more steps...

Step 3.2.2.1

Apply the distributive property.

$\frac{1 (1 - 1.7320508 i) + 2 i \sqrt{3} (1 - 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.2.2

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- 1.7320508 i) + 2 i \sqrt{3} (1 - 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.2.3

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- 1.7320508 i) + 2 i \sqrt{3} \cdot 1 + 2 i \sqrt{3} (- 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3

Simplify and combine like terms.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Multiply $1$ by $1$ .

$\frac{1 + 1 (- 1.7320508 i) + 2 i \sqrt{3} \cdot 1 + 2 i \sqrt{3} (- 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.2

Multiply $- 1.7320508 i$ by $1$ .

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} \cdot 1 + 2 i \sqrt{3} (- 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.3

Multiply $2$ by $1$ .

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} + 2 i \sqrt{3} (- 1.7320508 i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.4

Multiply $2 i \sqrt{3} (- 1.7320508 i)$ .

Tap for more steps...

Step 3.2.3.1.4.1

Multiply $- 1.7320508$ by $2$ .

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} - 3.46410161 i \sqrt{3} i}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.4.2

Multiply $\sqrt{3}$ by $- 3.46410161$ .

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} - 6 i i}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.4.3

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

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} - 6 (i^{1} i)}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.4.4

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

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} - 6 (i^{1} i^{1})}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.4.5

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

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} - 6 i^{1 + 1}}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.4.6

Add $1$ and $1$ .

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} - 6 i^{2}}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.5

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

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} - 6 \cdot - 1}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.1.6

Multiply $- 6$ by $- 1$ .

$\frac{1 - 1.7320508 i + 2 i \sqrt{3} + 6}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.2.3.2

Add $1$ and $6$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)}$

Step 3.3

Simplify the denominator.

Tap for more steps...

Step 3.3.1

Expand $(1 + 1.7320508 \cdot i) (1 - 1.7320508 i)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.1

Apply the distributive property.

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 (1 - 1.7320508 i) + 1.7320508 \cdot i (1 - 1.7320508 i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 \cdot 1 + 1 (- 1.7320508 i) + 1.7320508 \cdot i (1 - 1.7320508 i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 \cdot 1 + 1 (- 1.7320508 i) + 1.7320508 \cdot i \cdot 1 + 1.7320508 \cdot i (- 1.7320508 i)}$

Step 3.3.2

Simplify.

Tap for more steps...

Step 3.3.2.1

Multiply $1$ by $1$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 + 1 (- 1.7320508 i) + 1.7320508 \cdot i \cdot 1 + 1.7320508 \cdot i (- 1.7320508 i)}$

Step 3.3.2.2

Multiply $- 1.7320508$ by $1$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 - 1.7320508 i + 1.7320508 \cdot i \cdot 1 + 1.7320508 \cdot i (- 1.7320508 i)}$

Step 3.3.2.3

Multiply $1.7320508$ by $1$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 - 1.7320508 i + 1.7320508 \cdot i + 1.7320508 \cdot i (- 1.7320508 i)}$

Step 3.3.2.4

Multiply $- 1.7320508$ by $1.7320508$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 - 1.7320508 i + 1.7320508 \cdot i - 3 \cdot i i}$

Step 3.3.2.5

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

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 - 1.7320508 i + 1.7320508 \cdot i - 3 \cdot (i^{1} i)}$

Step 3.3.2.6

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

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 - 1.7320508 i + 1.7320508 \cdot i - 3 \cdot (i^{1} i^{1})}$

Step 3.3.2.7

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

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 - 1.7320508 i + 1.7320508 \cdot i - 3 \cdot i^{1 + 1}}$

Step 3.3.2.8

Add $1$ and $1$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 - 1.7320508 i + 1.7320508 \cdot i - 3 \cdot i^{2}}$

Step 3.3.2.9

Add $- 1.7320508 i$ and $1.7320508 \cdot i$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 + 0 i - 3 \cdot i^{2}}$

Step 3.3.3

Simplify each term.

Tap for more steps...

Step 3.3.3.1

Multiply $0$ by $i$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 + 0 - 3 \cdot i^{2}}$

Step 3.3.3.2

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

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 + 0 - 3 \cdot - 1}$

Step 3.3.3.3

Multiply $- 3$ by $- 1$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 + 0 + 3}$

Step 3.3.4

Add $1$ and $0$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1 + 3}$

Step 3.3.5

Add $1$ and $3$ .

$\frac{7 - 1.7320508 i + 2 i \sqrt{3}}{4}$

Step 4

Rewrite $7$ as $1 (7)$ .

$\frac{1 (7) - 1.7320508 i + 2 i \sqrt{3}}{4}$

Step 5

Factor $1$ out of $- 1.7320508 i$ .

$\frac{1 (7) + 1 (- 1.7320508 i) + 2 i \sqrt{3}}{4}$

Step 6

Factor $1$ out of $1 (7) + 1 (- 1.7320508 i)$ .

$\frac{1 (7 - 1.7320508 i) + 2 i \sqrt{3}}{4}$

Step 7

Factor $1$ out of $2 i \sqrt{3}$ .

$\frac{1 (7 - 1.7320508 i) + 1 (2 i \sqrt{3})}{4}$

Step 8

Factor $1$ out of $1 (7 - 1.7320508 i) + 1 (2 i \sqrt{3})$ .

$\frac{1 (7 - 1.7320508 i + 2 i \sqrt{3})}{4}$

Step 9

Factor $4$ out of $4$ .

$\frac{1 (7 - 1.7320508 i + 2 i \sqrt{3})}{4 (1)}$

Step 10

Separate fractions.

$\frac{1}{4} \cdot \frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1}$

Step 11

Simplify the expression.

Tap for more steps...

Step 11.1

Divide $1$ by $4$ .

$0.25 \frac{7 - 1.7320508 i + 2 i \sqrt{3}}{1}$

Step 11.2

Divide $7 - 1.7320508 i + 2 i \sqrt{3}$ by $1$ .

$0.25 (7 - 1.7320508 i + 2 i \sqrt{3})$

Step 12

Apply the distributive property.

$0.25 \cdot 7 + 0.25 (- 1.7320508 i) + 0.25 (2 i \sqrt{3})$

Step 13

Simplify.

Tap for more steps...

Step 13.1

Multiply $0.25$ by $7$ .

$1.75 + 0.25 (- 1.7320508 i) + 0.25 (2 i \sqrt{3})$

Step 13.2

Multiply $- 1.7320508$ by $0.25$ .

$1.75 - 0.4330127 i + 0.25 (2 i \sqrt{3})$

Step 13.3

Multiply $0.25 (2 i \sqrt{3})$ .

Tap for more steps...

Step 13.3.1

Multiply $2$ by $0.25$ .

$1.75 - 0.4330127 i + 0.5 (i \sqrt{3})$

Step 13.3.2

Multiply $\sqrt{3}$ by $0.5$ .

$1.75 - 0.4330127 i + 0.8660254 i$

Step 14

Add $- 0.4330127 i$ and $0.8660254 i$ .

$1.75 + 0.4330127 i$