Basic Math Examples

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

Step 1

Factor out $i^{2}$ .

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

Step 2

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 4

Expand $(2 - i) (2 - i)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $2$ by $2$ .

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

Step 5.1.2

Multiply $- 1$ by $2$ .

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

Step 5.1.3

Multiply $2$ by $- 1$ .

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

Step 5.1.4

Multiply $- i (- i)$ .

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Step 5.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.4.2

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

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

Step 5.1.4.3

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

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

Step 5.1.4.4

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

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

Step 5.1.4.5

Add $1$ and $1$ .

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

Step 5.1.4.6

Multiply $i^{2}$ by $1$ .

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

Step 5.1.5

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

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

Step 5.2

Subtract $1$ from $4$ .

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

Step 5.3

Subtract $2 i$ from $- 2 i$ .

$- \frac{1}{i (3 - 4 i)}$

Step 6

Apply the distributive property.

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

Step 7

Move $3$ to the left of $i$ .

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

Step 8

Multiply $i (- 4 i)$ .

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Step 8.1

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

$- \frac{1}{3 \cdot i - 4 (i^{1} i)}$

Step 8.2

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

$- \frac{1}{3 \cdot i - 4 (i^{1} i^{1})}$

Step 8.3

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

$- \frac{1}{3 \cdot i - 4 i^{1 + 1}}$

Step 8.4

Add $1$ and $1$ .

$- \frac{1}{3 \cdot i - 4 i^{2}}$

Step 9

Simplify each term.

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Step 9.1

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

$- \frac{1}{3 i - 4 \cdot - 1}$

Step 9.2

Multiply $- 4$ by $- 1$ .

$- \frac{1}{3 i + 4}$

Step 10

Reorder $3 i$ and $4$ .

$- \frac{1}{4 + 3 i}$

Step 11

Multiply the numerator and denominator of $\frac{1}{4 + 3 i}$ by the conjugate of $4 + 3 i$ to make the denominator real.

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

Step 12

Multiply.

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Step 12.1

Combine.

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

Step 12.2

Multiply $4 - 3 i$ by $1$ .

$- \frac{4 - 3 i}{(4 + 3 i) (4 - 3 i)}$

Step 12.3

Simplify the denominator.

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Step 12.3.1

Expand $(4 + 3 i) (4 - 3 i)$ using the FOIL Method.

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Step 12.3.1.1

Apply the distributive property.

$- \frac{4 - 3 i}{4 (4 - 3 i) + 3 i (4 - 3 i)}$

Step 12.3.1.2

Apply the distributive property.

$- \frac{4 - 3 i}{4 \cdot 4 + 4 (- 3 i) + 3 i (4 - 3 i)}$

Step 12.3.1.3

Apply the distributive property.

$- \frac{4 - 3 i}{4 \cdot 4 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Step 12.3.2

Simplify.

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Step 12.3.2.1

Multiply $4$ by $4$ .

$- \frac{4 - 3 i}{16 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Step 12.3.2.2

Multiply $- 3$ by $4$ .

$- \frac{4 - 3 i}{16 - 12 i + 3 i \cdot 4 + 3 i (- 3 i)}$

Step 12.3.2.3

Multiply $4$ by $3$ .

$- \frac{4 - 3 i}{16 - 12 i + 12 i + 3 i (- 3 i)}$

Step 12.3.2.4

Multiply $- 3$ by $3$ .

$- \frac{4 - 3 i}{16 - 12 i + 12 i - 9 i i}$

Step 12.3.2.5

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

$- \frac{4 - 3 i}{16 - 12 i + 12 i - 9 (i^{1} i)}$

Step 12.3.2.6

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

$- \frac{4 - 3 i}{16 - 12 i + 12 i - 9 (i^{1} i^{1})}$

Step 12.3.2.7

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

$- \frac{4 - 3 i}{16 - 12 i + 12 i - 9 i^{1 + 1}}$

Step 12.3.2.8

Add $1$ and $1$ .

$- \frac{4 - 3 i}{16 - 12 i + 12 i - 9 i^{2}}$

Step 12.3.2.9

Add $- 12 i$ and $12 i$ .

$- \frac{4 - 3 i}{16 + 0 - 9 i^{2}}$

Step 12.3.2.10

Add $16$ and $0$ .

$- \frac{4 - 3 i}{16 - 9 i^{2}}$

Step 12.3.3

Simplify each term.

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Step 12.3.3.1

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

$- \frac{4 - 3 i}{16 - 9 \cdot - 1}$

Step 12.3.3.2

Multiply $- 9$ by $- 1$ .

$- \frac{4 - 3 i}{16 + 9}$

Step 12.3.4

Add $16$ and $9$ .

$- \frac{4 - 3 i}{25}$

Step 13

Split the fraction $\frac{4 - 3 i}{25}$ into two fractions.

$- (\frac{4}{25} + \frac{- 3 i}{25})$

Step 14

Move the negative in front of the fraction.

$- (\frac{4}{25} - \frac{3 i}{25})$

Step 15

Apply the distributive property.

$- \frac{4}{25} - - \frac{3 i}{25}$

Step 16

Multiply $- - \frac{3 i}{25}$ .

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Step 16.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{25} + 1 \frac{3 i}{25}$

Step 16.2

Multiply $\frac{3 i}{25}$ by $1$ .

$- \frac{4}{25} + \frac{3 i}{25}$