Basic Math Examples

$\frac{9 + i}{i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator and denominator of $\frac{9 + i}{i}$ by the conjugate of $i$ to make the denominator real.

$\frac{9 + i}{i} \cdot \frac{i}{i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2

Multiply.

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Step 1.2.1

Combine.

$\frac{(9 + i) i}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Apply the distributive property.

$\frac{9 i + i i}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.2.2

Multiply $i i$ .

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Step 1.2.2.2.1

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

$\frac{9 i + i^{1} i}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.2.2.2

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

$\frac{9 i + i^{1} i^{1}}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.2.2.3

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

$\frac{9 i + i^{1 + 1}}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.2.2.4

Add $1$ and $1$ .

$\frac{9 i + i^{2}}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.2.3

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

$\frac{9 i - 1}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.2.4

Reorder $9 i$ and $- 1$ .

$\frac{- 1 + 9 i}{i i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

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

$\frac{- 1 + 9 i}{i^{1} i} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.3.2

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

$\frac{- 1 + 9 i}{i^{1} i^{1}} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.3.3

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

$\frac{- 1 + 9 i}{i^{1 + 1}} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.3.4

Add $1$ and $1$ .

$\frac{- 1 + 9 i}{i^{2}} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.2.3.5

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

$\frac{- 1 + 9 i}{- 1} - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.3

Move the negative one from the denominator of $\frac{- 1 + 9 i}{- 1}$ .

$- 1 \cdot (- 1 + 9 i) - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.4

Rewrite $- 1 \cdot (- 1 + 9 i)$ as $- (- 1 + 9 i)$ .

$- (- 1 + 9 i) - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.5

Apply the distributive property.

$- - 1 - (9 i) - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.6

Multiply $- 1$ by $- 1$ .

$1 - (9 i) - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.7

Multiply $9$ by $- 1$ .

$1 - 9 i - {(\frac{5 - 5 i}{2 + i})}^{2}$

Step 1.8

Multiply the numerator and denominator of $\frac{5 - 5 i}{2 + 1 i}$ by the conjugate of $2 + 1 i$ to make the denominator real.

$1 - 9 i - {(\frac{5 - 5 i}{2 + 1 i} \cdot \frac{2 - i}{2 - i})}^{2}$

Step 1.9

Multiply.

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Step 1.9.1

Combine.

$1 - 9 i - {(\frac{(5 - 5 i) (2 - i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2

Simplify the numerator.

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Step 1.9.2.1

Expand $(5 - 5 i) (2 - i)$ using the FOIL Method.

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Step 1.9.2.1.1

Apply the distributive property.

$1 - 9 i - {(\frac{5 (2 - i) - 5 i (2 - i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.1.2

Apply the distributive property.

$1 - 9 i - {(\frac{5 \cdot 2 + 5 (- i) - 5 i (2 - i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.1.3

Apply the distributive property.

$1 - 9 i - {(\frac{5 \cdot 2 + 5 (- i) - 5 i \cdot 2 - 5 i (- i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2

Simplify and combine like terms.

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Step 1.9.2.2.1

Simplify each term.

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Step 1.9.2.2.1.1

Multiply $5$ by $2$ .

$1 - 9 i - {(\frac{10 + 5 (- i) - 5 i \cdot 2 - 5 i (- i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.2

Multiply $- 1$ by $5$ .

$1 - 9 i - {(\frac{10 - 5 i - 5 i \cdot 2 - 5 i (- i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.3

Multiply $2$ by $- 5$ .

$1 - 9 i - {(\frac{10 - 5 i - 10 i - 5 i (- i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.4

Multiply $- 5 i (- i)$ .

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Step 1.9.2.2.1.4.1

Multiply $- 1$ by $- 5$ .

$1 - 9 i - {(\frac{10 - 5 i - 10 i + 5 i i}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.4.2

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

$1 - 9 i - {(\frac{10 - 5 i - 10 i + 5 (i^{1} i)}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.4.3

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

$1 - 9 i - {(\frac{10 - 5 i - 10 i + 5 (i^{1} i^{1})}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.4.4

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

$1 - 9 i - {(\frac{10 - 5 i - 10 i + 5 i^{1 + 1}}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.4.5

Add $1$ and $1$ .

$1 - 9 i - {(\frac{10 - 5 i - 10 i + 5 i^{2}}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.5

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

$1 - 9 i - {(\frac{10 - 5 i - 10 i + 5 \cdot - 1}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.1.6

Multiply $5$ by $- 1$ .

$1 - 9 i - {(\frac{10 - 5 i - 10 i - 5}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.2

Subtract $5$ from $10$ .

$1 - 9 i - {(\frac{5 - 5 i - 10 i}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.2.2.3

Subtract $10 i$ from $- 5 i$ .

$1 - 9 i - {(\frac{5 - 15 i}{(2 + 1 i) (2 - i)})}^{2}$

Step 1.9.3

Simplify the denominator.

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Step 1.9.3.1

Expand $(2 + 1 i) (2 - i)$ using the FOIL Method.

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Step 1.9.3.1.1

Apply the distributive property.

$1 - 9 i - {(\frac{5 - 15 i}{2 (2 - i) + 1 i (2 - i)})}^{2}$

Step 1.9.3.1.2

Apply the distributive property.

$1 - 9 i - {(\frac{5 - 15 i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)})}^{2}$

Step 1.9.3.1.3

Apply the distributive property.

$1 - 9 i - {(\frac{5 - 15 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)})}^{2}$

Step 1.9.3.2

Simplify.

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Step 1.9.3.2.1

Multiply $2$ by $2$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)})}^{2}$

Step 1.9.3.2.2

Multiply $- 1$ by $2$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)})}^{2}$

Step 1.9.3.2.3

Multiply $2$ by $1$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 - 2 i + 2 i + 1 i (- i)})}^{2}$

Step 1.9.3.2.4

Multiply $- 1$ by $1$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 - 2 i + 2 i - i i})}^{2}$

Step 1.9.3.2.5

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

$1 - 9 i - {(\frac{5 - 15 i}{4 - 2 i + 2 i - (i^{1} i)})}^{2}$

Step 1.9.3.2.6

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

$1 - 9 i - {(\frac{5 - 15 i}{4 - 2 i + 2 i - (i^{1} i^{1})})}^{2}$

Step 1.9.3.2.7

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

$1 - 9 i - {(\frac{5 - 15 i}{4 - 2 i + 2 i - i^{1 + 1}})}^{2}$

Step 1.9.3.2.8

Add $1$ and $1$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 - 2 i + 2 i - i^{2}})}^{2}$

Step 1.9.3.2.9

Add $- 2 i$ and $2 i$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 + 0 - i^{2}})}^{2}$

Step 1.9.3.2.10

Add $4$ and $0$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 - i^{2}})}^{2}$

Step 1.9.3.3

Simplify each term.

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Step 1.9.3.3.1

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

$1 - 9 i - {(\frac{5 - 15 i}{4 - - 1})}^{2}$

Step 1.9.3.3.2

Multiply $- 1$ by $- 1$ .

$1 - 9 i - {(\frac{5 - 15 i}{4 + 1})}^{2}$

Step 1.9.3.4

Add $4$ and $1$ .

$1 - 9 i - {(\frac{5 - 15 i}{5})}^{2}$

Step 1.10

Cancel the common factor of $5 - 15 i$ and $5$ .

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Step 1.10.1

Factor $5$ out of $5$ .

$1 - 9 i - {(\frac{5 \cdot 1 - 15 i}{5})}^{2}$

Step 1.10.2

Factor $5$ out of $- 15 i$ .

$1 - 9 i - {(\frac{5 \cdot 1 + 5 (- 3 i)}{5})}^{2}$

Step 1.10.3

Factor $5$ out of $5 (1) + 5 (- 3 i)$ .

$1 - 9 i - {(\frac{5 (1 - 3 i)}{5})}^{2}$

Step 1.10.4

Cancel the common factors.

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Step 1.10.4.1

Factor $5$ out of $5$ .

$1 - 9 i - {(\frac{5 (1 - 3 i)}{5 (1)})}^{2}$

Step 1.10.4.2

Cancel the common factor.

$1 - 9 i - {(\frac{5 (1 - 3 i)}{5 \cdot 1})}^{2}$

Step 1.10.4.3

Rewrite the expression.

$1 - 9 i - {(\frac{1 - 3 i}{1})}^{2}$

Step 1.10.4.4

Divide $1 - 3 i$ by $1$ .

$1 - 9 i - {(1 - 3 i)}^{2}$

Step 1.11

Rewrite ${(1 - 3 i)}^{2}$ as $(1 - 3 i) (1 - 3 i)$ .

$1 - 9 i - ((1 - 3 i) (1 - 3 i))$

Step 1.12

Expand $(1 - 3 i) (1 - 3 i)$ using the FOIL Method.

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Step 1.12.1

Apply the distributive property.

$1 - 9 i - (1 (1 - 3 i) - 3 i (1 - 3 i))$

Step 1.12.2

Apply the distributive property.

$1 - 9 i - (1 \cdot 1 + 1 (- 3 i) - 3 i (1 - 3 i))$

Step 1.12.3

Apply the distributive property.

$1 - 9 i - (1 \cdot 1 + 1 (- 3 i) - 3 i \cdot 1 - 3 i (- 3 i))$

Step 1.13

Simplify and combine like terms.

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Step 1.13.1

Simplify each term.

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Step 1.13.1.1

Multiply $1$ by $1$ .

$1 - 9 i - (1 + 1 (- 3 i) - 3 i \cdot 1 - 3 i (- 3 i))$

Step 1.13.1.2

Multiply $- 3 i$ by $1$ .

$1 - 9 i - (1 - 3 i - 3 i \cdot 1 - 3 i (- 3 i))$

Step 1.13.1.3

Multiply $- 3$ by $1$ .

$1 - 9 i - (1 - 3 i - 3 i - 3 i (- 3 i))$

Step 1.13.1.4

Multiply $- 3 i (- 3 i)$ .

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Step 1.13.1.4.1

Multiply $- 3$ by $- 3$ .

$1 - 9 i - (1 - 3 i - 3 i + 9 i i)$

Step 1.13.1.4.2

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

$1 - 9 i - (1 - 3 i - 3 i + 9 (i^{1} i))$

Step 1.13.1.4.3

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

$1 - 9 i - (1 - 3 i - 3 i + 9 (i^{1} i^{1}))$

Step 1.13.1.4.4

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

$1 - 9 i - (1 - 3 i - 3 i + 9 i^{1 + 1})$

Step 1.13.1.4.5

Add $1$ and $1$ .

$1 - 9 i - (1 - 3 i - 3 i + 9 i^{2})$

Step 1.13.1.5

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

$1 - 9 i - (1 - 3 i - 3 i + 9 \cdot - 1)$

Step 1.13.1.6

Multiply $9$ by $- 1$ .

$1 - 9 i - (1 - 3 i - 3 i - 9)$

Step 1.13.2

Subtract $9$ from $1$ .

$1 - 9 i - (- 8 - 3 i - 3 i)$

Step 1.13.3

Subtract $3 i$ from $- 3 i$ .

$1 - 9 i - (- 8 - 6 i)$

Step 1.14

Apply the distributive property.

$1 - 9 i - - 8 - (- 6 i)$

Step 1.15

Multiply $- 1$ by $- 8$ .

$1 - 9 i + 8 - (- 6 i)$

Step 1.16

Multiply $- 6$ by $- 1$ .

$1 - 9 i + 8 + 6 i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $1$ and $8$ .

$9 - 9 i + 6 i$

Step 2.2

Add $- 9 i$ and $6 i$ .

$9 - 3 i$