Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $i i$ .

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

Add $1$ and $1$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Move $- 1$ to the left of $i$ .

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

Step 1.3.1.4

Rewrite $- 1 i$ as $- i$ .

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

Step 1.3.1.5

Rewrite $- 1 i$ as $- i$ .

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

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.3.2

Add $- 1$ and $1$ .

$0 - i - i + \frac{2}{{(i - 1)}^{2}}$

Step 1.3.3

Subtract $i$ from $0$ .

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

Step 1.3.4

Subtract $i$ from $- i$ .

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $i i$ .

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Step 1.6.1.1.1

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

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

Step 1.6.1.1.2

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

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

Step 1.6.1.1.3

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

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

Step 1.6.1.1.4

Add $1$ and $1$ .

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

Step 1.6.1.2

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

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

Step 1.6.1.3

Move $- 1$ to the left of $i$ .

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

Step 1.6.1.4

Rewrite $- 1 i$ as $- i$ .

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

Step 1.6.1.5

Rewrite $- 1 i$ as $- i$ .

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

Step 1.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.6.2

Add $- 1$ and $1$ .

$- 2 i + \frac{2}{0 - i - i}$

Step 1.6.3

Subtract $i$ from $0$ .

$- 2 i + \frac{2}{- i - i}$

Step 1.6.4

Subtract $i$ from $- i$ .

$- 2 i + \frac{2}{- 2 i}$

Step 1.7

Cancel the common factor of $2$ and $- 2$ .

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Step 1.7.1

Factor $2$ out of $2$ .

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

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $2$ out of $- 2 i$ .

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

Step 1.7.2.2

Cancel the common factor.

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

Step 1.7.2.3

Rewrite the expression.

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

Step 1.8

Cancel the common factor of $1$ and $- 1$ .

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Step 1.8.1

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

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

Step 1.8.2

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

Multiply.

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Step 1.10.1

Combine.

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

Step 1.10.2

Multiply $i$ by $1$ .

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

Step 1.10.3

Simplify the denominator.

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Step 1.10.3.1

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

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

Step 1.10.3.2

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

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

Step 1.10.3.3

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

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

Step 1.10.3.4

Add $1$ and $1$ .

$- 2 i - \frac{i}{i^{2}}$

Step 1.10.3.5

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

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

Step 1.11

Move the negative one from the denominator of $\frac{i}{- 1}$ .

$- 2 i - (- 1 \cdot i)$

Step 1.12

Rewrite $- 1 \cdot i$ as $- i$ .

$- 2 i - - i$

Step 1.13

Multiply $- 1$ by $- 1$ .

$- 2 i + 1 i$

Step 1.14

Multiply $i$ by $1$ .

$- 2 i + i$

Step 2

Add $- 2 i$ and $i$ .

$- i$