Algebra Examples

$(6 + y i) (5 + i) - (6 + y i) (5 - 4 i)$

Step 1

Simplify each term.

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Step 1.1

Expand $(6 + y i) (5 + i)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$6 (5 + i) + y i (5 + i) - (6 + y i) (5 - 4 i)$

Step 1.1.2

Apply the distributive property.

$6 \cdot 5 + 6 i + y i (5 + i) - (6 + y i) (5 - 4 i)$

Step 1.1.3

Apply the distributive property.

$6 \cdot 5 + 6 i + y i \cdot 5 + y i i - (6 + y i) (5 - 4 i)$

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply $6$ by $5$ .

$30 + 6 i + y i \cdot 5 + y i i - (6 + y i) (5 - 4 i)$

Step 1.2.2

Move $5$ to the left of $y i$ .

$30 + 6 i + 5 \cdot (y i) + y i i - (6 + y i) (5 - 4 i)$

Step 1.2.3

Multiply $y i i$ .

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Step 1.2.3.1

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

$30 + 6 i + 5 y i + y (i^{1} i) - (6 + y i) (5 - 4 i)$

Step 1.2.3.2

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

$30 + 6 i + 5 y i + y (i^{1} i^{1}) - (6 + y i) (5 - 4 i)$

Step 1.2.3.3

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

$30 + 6 i + 5 y i + y i^{1 + 1} - (6 + y i) (5 - 4 i)$

Step 1.2.3.4

Add $1$ and $1$ .

$30 + 6 i + 5 y i + y i^{2} - (6 + y i) (5 - 4 i)$

Step 1.2.4

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

$30 + 6 i + 5 y i + y \cdot - 1 - (6 + y i) (5 - 4 i)$

Step 1.2.5

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

$30 + 6 i + 5 y i - 1 \cdot y - (6 + y i) (5 - 4 i)$

Step 1.2.6

Rewrite $- 1 y$ as $- y$ .

$30 + 6 i + 5 y i - y - (6 + y i) (5 - 4 i)$

Step 1.3

Apply the distributive property.

$30 + 6 i + 5 y i - y + (- 1 \cdot 6 - (y i)) (5 - 4 i)$

Step 1.4

Multiply $- 1$ by $6$ .

$30 + 6 i + 5 y i - y + (- 6 - (y i)) (5 - 4 i)$

Step 1.5

Expand $(- 6 - y i) (5 - 4 i)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$30 + 6 i + 5 y i - y - 6 (5 - 4 i) - y i (5 - 4 i)$

Step 1.5.2

Apply the distributive property.

$30 + 6 i + 5 y i - y - 6 \cdot 5 - 6 (- 4 i) - y i (5 - 4 i)$

Step 1.5.3

Apply the distributive property.

$30 + 6 i + 5 y i - y - 6 \cdot 5 - 6 (- 4 i) - y i \cdot 5 - y i (- 4 i)$

Step 1.6

Simplify each term.

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Step 1.6.1

Multiply $- 6$ by $5$ .

$30 + 6 i + 5 y i - y - 30 - 6 (- 4 i) - y i \cdot 5 - y i (- 4 i)$

Step 1.6.2

Multiply $- 4$ by $- 6$ .

$30 + 6 i + 5 y i - y - 30 + 24 i - y i \cdot 5 - y i (- 4 i)$

Step 1.6.3

Multiply $5$ by $- 1$ .

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i - y i (- 4 i)$

Step 1.6.4

Multiply $- y i (- 4 i)$ .

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Step 1.6.4.1

Multiply $- 4$ by $- 1$ .

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i + 4 y i i$

Step 1.6.4.2

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

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i + 4 y (i^{1} i)$

Step 1.6.4.3

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

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i + 4 y (i^{1} i^{1})$

Step 1.6.4.4

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

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i + 4 y i^{1 + 1}$

Step 1.6.4.5

Add $1$ and $1$ .

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i + 4 y i^{2}$

Step 1.6.5

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

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i + 4 y \cdot - 1$

Step 1.6.6

Multiply $- 1$ by $4$ .

$30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i - 4 y$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $30 + 6 i + 5 y i - y - 30 + 24 i - 5 y i - 4 y$ .

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Step 2.1.1

Subtract $30$ from $30$ .

$6 i + 5 y i - y + 0 + 24 i - 5 y i - 4 y$

Step 2.1.2

Add $6 i + 5 y i - y$ and $0$ .

$6 i + 5 y i - y + 24 i - 5 y i - 4 y$

Step 2.1.3

Subtract $5 y i$ from $5 y i$ .

$6 i + 0 - y + 24 i - 4 y$

Step 2.1.4

Add $6 i$ and $0$ .

$6 i - y + 24 i - 4 y$

Step 2.2

Add $6 i$ and $24 i$ .

$- y + 30 i - 4 y$

Step 2.3

Subtract $4 y$ from $- y$ .

$- 5 y + 30 i$