Algebra Examples

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

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Apply the distributive property.

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

Step 1.2

Multiply $2$ by $1$ .

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

Step 1.3

Multiply $- 1$ by $2$ .

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

Step 1.4

Subtract $3$ from $2$ .

$\frac{- 2 i \sqrt{2} - 1}{{(1 - i \sqrt{2})}^{2}}$

Step 2

Rewrite ${(1 - i \sqrt{2})}^{2}$ as $(1 - i \sqrt{2}) (1 - i \sqrt{2})$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $1$ by $1$ .

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

Step 4.1.2

Multiply $- i \sqrt{2}$ by $1$ .

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

Step 4.1.3

Multiply $- 1$ by $1$ .

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

Step 4.1.4

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

Tap for more steps...

Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.4.2

Multiply $\sqrt{2}$ by $1$ .

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

Step 4.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + {\sqrt{2}}^{1} \sqrt{2} i i}$

Step 4.1.4.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + {\sqrt{2}}^{1} {\sqrt{2}}^{1} i i}$

Step 4.1.4.5

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

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + {\sqrt{2}}^{1 + 1} i i}$

Step 4.1.4.6

Add $1$ and $1$ .

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + {\sqrt{2}}^{2} i i}$

Step 4.1.4.7

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

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

Step 4.1.4.8

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

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

Step 4.1.4.9

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

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + {\sqrt{2}}^{2} i^{1 + 1}}$

Step 4.1.4.10

Add $1$ and $1$ .

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + {\sqrt{2}}^{2} i^{2}}$

Step 4.1.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 4.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + 2^{\frac{2}{2}} i^{2}}$

Step 4.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.5.4.1

Cancel the common factor.

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + 2^{\frac{2}{2}} i^{2}}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + 2^{1} i^{2}}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} + 2 i^{2}}$

Step 4.1.6

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

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

Step 4.1.7

Multiply $2$ by $- 1$ .

$\frac{- 2 i \sqrt{2} - 1}{1 - i \sqrt{2} - i \sqrt{2} - 2}$

Step 4.2

Subtract $2$ from $1$ .

$\frac{- 2 i \sqrt{2} - 1}{- i \sqrt{2} - i \sqrt{2} - 1}$

Step 4.3

Subtract $i \sqrt{2}$ from $- i \sqrt{2}$ .

$\frac{- 2 i \sqrt{2} - 1}{- 2 i \sqrt{2} - 1}$

Step 5

Reorder $- 2 i \sqrt{2}$ and $- 1$ .

$\frac{- 2 i \sqrt{2} - 1}{- 1 - 2 i \sqrt{2}}$

Step 6

Cancel the common factor of $- 2 i \sqrt{2} - 1$ and $- 1 - 2 i \sqrt{2}$ .

Tap for more steps...

Step 6.1

Reorder terms.

$\frac{- 2 i \sqrt{2} - 1}{- 2 i \sqrt{2} - 1}$

Step 6.2

Cancel the common factor.

$\frac{- 2 i \sqrt{2} - 1}{- 2 i \sqrt{2} - 1}$

Step 6.3

Rewrite the expression.

$1$