Trigonometry Examples

$(3 i + j) \cdot (i + 3 j)$

Step 1

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

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Step 1.1

Apply the distributive property.

$3 i (i + 3 j) + j (i + 3 j)$

Step 1.2

Apply the distributive property.

$3 i i + 3 i (3 j) + j (i + 3 j)$

Step 1.3

Apply the distributive property.

$3 i i + 3 i (3 j) + j i + j (3 j)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $3 i i$ .

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Step 2.1.1.1

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

$3 (i^{1} i) + 3 i (3 j) + j i + j (3 j)$

Step 2.1.1.2

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

$3 (i^{1} i^{1}) + 3 i (3 j) + j i + j (3 j)$

Step 2.1.1.3

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

$3 i^{1 + 1} + 3 i (3 j) + j i + j (3 j)$

Step 2.1.1.4

Add $1$ and $1$ .

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

Step 2.1.2

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

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

Step 2.1.3

Multiply $3$ by $- 1$ .

$- 3 + 3 i (3 j) + j i + j (3 j)$

Step 2.1.4

Multiply $3$ by $3$ .

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

Step 2.1.5

Rewrite using the commutative property of multiplication.

$- 3 + 9 i j + j i + 3 j \cdot j$

Step 2.1.6

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

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Step 2.1.6.1

Move $j$ .

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

Step 2.1.6.2

Multiply $j$ by $j$ .

$- 3 + 9 i j + j i + 3 j^{2}$

Step 2.2

Add $9 i j$ and $j i$ .

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Step 2.2.1

Move $i$ .

$- 3 + 9 j i + j i + 3 j^{2}$

Step 2.2.2

Add $9 j i$ and $j i$ .

$- 3 + 10 j i + 3 j^{2}$