Algebra Examples

(a + b i) (a - b i)

$(a + b i) (a - b i)$

Step 1

Expand

(a + b i) (a - b i)

$(a + b i) (a - b i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

a (a - b i) + b i (a - b i)

$a (a - b i) + b i (a - b i)$

Step 1.2

Apply the distributive property.

a \cdot a + a (- b i) + b i (a - b i)

$a \cdot a + a (- b i) + b i (a - b i)$

Step 1.3

Apply the distributive property.

a \cdot a + a (- b i) + b i a + b i (- b i)

$a \cdot a + a (- b i) + b i a + b i (- b i)$

a \cdot a + a (- b i) + b i a + b i (- b i)

$a \cdot a + a (- b i) + b i a + b i (- b i)$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in

a \cdot a + a (- b i) + b i a + b i (- b i)

$a \cdot a + a (- b i) + b i a + b i (- b i)$ .

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Step 2.1.1

Reorder the factors in the terms

a (- b i)

$a (- b i)$ and

b i a

$b i a$ .

a \cdot a - i a b + i a b + b i (- b i)

$a \cdot a - i a b + i a b + b i (- b i)$

Step 2.1.2

Add

- i a b

$- i a b$ and

i a b

$i a b$ .

a \cdot a + 0 + b i (- b i)

$a \cdot a + 0 + b i (- b i)$

Step 2.1.3

Add

a \cdot a

$a \cdot a$ and

0

$0$ .

a \cdot a + b i (- b i)

$a \cdot a + b i (- b i)$

a \cdot a + b i (- b i)

$a \cdot a + b i (- b i)$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply

a

$a$ by

a

$a$ .

a^{2} + b i (- b i)

$a^{2} + b i (- b i)$

Step 2.2.2

Multiply

b

$b$ by

b

$b$ by adding the exponents.

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Step 2.2.2.1

Move

b

$b$ .

a^{2} + b \cdot b i (- 1 i)

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

Step 2.2.2.2

Multiply

b

$b$ by

b

$b$ .

a^{2} + b^{2} i (- 1 i)

$a^{2} + b^{2} i (- 1 i)$

a^{2} + b^{2} i (- 1 i)

$a^{2} + b^{2} i (- 1 i)$

Step 2.2.3

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

a^{2} + b^{2} i (- i)

$a^{2} + b^{2} i (- i)$

Step 2.2.4

Multiply

b^{2} i (- i)

$b^{2} i (- i)$ .

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Step 2.2.4.1

Raise

i

$i$ to the power of

1

$1$ .

a^{2} + b^{2} (- (i^{1} i))

$a^{2} + b^{2} (- (i^{1} i))$

Step 2.2.4.2

Raise

i

$i$ to the power of

1

$1$ .

a^{2} + b^{2} (- (i^{1} i^{1}))

$a^{2} + b^{2} (- (i^{1} i^{1}))$

Step 2.2.4.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

a^{2} + b^{2} (- i^{1 + 1})

$a^{2} + b^{2} (- i^{1 + 1})$

Step 2.2.4.4

Add

1

$1$ and

1

$1$ .

a^{2} + b^{2} (- i^{2})

$a^{2} + b^{2} (- i^{2})$

a^{2} + b^{2} (- i^{2})

$a^{2} + b^{2} (- i^{2})$

Step 2.2.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

a^{2} + b^{2} (- - 1)

$a^{2} + b^{2} (- - 1)$

Step 2.2.6

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

a^{2} + b^{2} \cdot 1

$a^{2} + b^{2} \cdot 1$

Step 2.2.7

Multiply

b^{2}

$b^{2}$ by

1

$1$ .

a^{2} + b^{2}

$a^{2} + b^{2}$

a^{2} + b^{2}

$a^{2} + b^{2}$

a^{2} + b^{2}

$a^{2} + b^{2}$

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$