Basic Math Examples

$(2 i + 3 j + 5 k) \cdot (i + 3 j + 2 k)$

Step 1

Expand $(2 i + 3 j + 5 k) (i + 3 j + 2 k)$ by multiplying each term in the first expression by each term in the second expression.

$2 i i + 2 i (3 j) + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2

Simplify each term.

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Step 2.1

Multiply $2 i i$ .

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Step 2.1.1

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

$2 (i^{1} i) + 2 i (3 j) + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.1.2

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

$2 (i^{1} i^{1}) + 2 i (3 j) + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.1.3

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

$2 i^{1 + 1} + 2 i (3 j) + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.1.4

Add $1$ and $1$ .

$2 i^{2} + 2 i (3 j) + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.2

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

$2 \cdot - 1 + 2 i (3 j) + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.3

Multiply $2$ by $- 1$ .

$- 2 + 2 i (3 j) + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.4

Multiply $3$ by $2$ .

$- 2 + 6 i j + 2 i (2 k) + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.5

Multiply $2$ by $2$ .

$- 2 + 6 i j + 4 i k + 3 j i + 3 j (3 j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.6

Rewrite using the commutative property of multiplication.

$- 2 + 6 i j + 4 i k + 3 j i + 3 \cdot 3 j \cdot j + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.7

Multiply $j$ by $j$ by adding the exponents.

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Step 2.7.1

Move $j$ .

$- 2 + 6 i j + 4 i k + 3 j i + 3 \cdot 3 (j \cdot j) + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.7.2

Multiply $j$ by $j$ .

$- 2 + 6 i j + 4 i k + 3 j i + 3 \cdot 3 j^{2} + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.8

Multiply $3$ by $3$ .

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 3 j (2 k) + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.9

Rewrite using the commutative property of multiplication.

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 3 \cdot 2 j k + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.10

Multiply $3$ by $2$ .

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 6 j k + 5 k i + 5 k (3 j) + 5 k (2 k)$

Step 2.11

Rewrite using the commutative property of multiplication.

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 6 j k + 5 k i + 5 \cdot 3 k j + 5 k (2 k)$

Step 2.12

Multiply $5$ by $3$ .

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 6 j k + 5 k i + 15 k j + 5 k (2 k)$

Step 2.13

Rewrite using the commutative property of multiplication.

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 6 j k + 5 k i + 15 k j + 5 \cdot 2 k \cdot k$

Step 2.14

Multiply $k$ by $k$ by adding the exponents.

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Step 2.14.1

Move $k$ .

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 6 j k + 5 k i + 15 k j + 5 \cdot 2 (k \cdot k)$

Step 2.14.2

Multiply $k$ by $k$ .

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 6 j k + 5 k i + 15 k j + 5 \cdot 2 k^{2}$

Step 2.15

Multiply $5$ by $2$ .

$- 2 + 6 i j + 4 i k + 3 j i + 9 j^{2} + 6 j k + 5 k i + 15 k j + 10 k^{2}$

Step 3

Add $6 i j$ and $3 j i$ .

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Step 3.1

Move $i$ .

$- 2 + 6 j i + 3 j i + 4 i k + 9 j^{2} + 6 j k + 5 k i + 15 k j + 10 k^{2}$

Step 3.2

Add $6 j i$ and $3 j i$ .

$- 2 + 9 j i + 4 i k + 9 j^{2} + 6 j k + 5 k i + 15 k j + 10 k^{2}$

Step 4

Add $4 i k$ and $5 k i$ .

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Step 4.1

Move $i$ .

$- 2 + 9 j i + 9 j^{2} + 6 j k + 4 k i + 5 k i + 15 k j + 10 k^{2}$

Step 4.2

Add $4 k i$ and $5 k i$ .

$- 2 + 9 j i + 9 j^{2} + 6 j k + 9 k i + 15 k j + 10 k^{2}$

Step 5

Add $6 j k$ and $15 k j$ .

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Step 5.1

Move $k$ .

$- 2 + 9 j i + 9 j^{2} + 6 j k + 15 j k + 9 k i + 10 k^{2}$

Step 5.2

Add $6 j k$ and $15 j k$ .

$- 2 + 9 j i + 9 j^{2} + 21 j k + 9 k i + 10 k^{2}$

Step 6

Move $- 2$ .

$9 j i + 9 j^{2} + 21 j k + 9 k i + 10 k^{2} - 2$

Step 7

Move $9 k i$ .

$9 j i + 9 j^{2} + 21 j k + 10 k^{2} + 9 k i - 2$

Step 8

Move $9 j i$ .

$9 j^{2} + 21 j k + 10 k^{2} + 9 j i + 9 k i - 2$