Simplify (3-5i)(2+9i)+(7+2i)(7-2i)

$(3 - 5 i) (2 + 9 i) + (7 + 2 i) (7 - 2 i)$

Step 1

Simplify each term.

Expand $(3 - 5 i) (2 + 9 i)$ using the FOIL Method.

Apply the distributive property.

$3 (2 + 9 i) - 5 i (2 + 9 i) + (7 + 2 i) (7 - 2 i)$

Apply the distributive property.

$3 \cdot 2 + 3 (9 i) - 5 i (2 + 9 i) + (7 + 2 i) (7 - 2 i)$

Apply the distributive property.

$3 \cdot 2 + 3 (9 i) - 5 i \cdot 2 - 5 i (9 i) + (7 + 2 i) (7 - 2 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $2$ .

$6 + 3 (9 i) - 5 i \cdot 2 - 5 i (9 i) + (7 + 2 i) (7 - 2 i)$

Multiply $9$ by $3$ .

$6 + 27 i - 5 i \cdot 2 - 5 i (9 i) + (7 + 2 i) (7 - 2 i)$

Multiply $2$ by $- 5$ .

$6 + 27 i - 10 i - 5 i (9 i) + (7 + 2 i) (7 - 2 i)$

Multiply $- 5 i (9 i)$ .

Multiply $9$ by $- 5$ .

$6 + 27 i - 10 i - 45 i i + (7 + 2 i) (7 - 2 i)$

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

$6 + 27 i - 10 i - 45 (i^{1} i) + (7 + 2 i) (7 - 2 i)$

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

$6 + 27 i - 10 i - 45 (i^{1} i^{1}) + (7 + 2 i) (7 - 2 i)$

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

$6 + 27 i - 10 i - 45 i^{1 + 1} + (7 + 2 i) (7 - 2 i)$

Add $1$ and $1$ .

$6 + 27 i - 10 i - 45 i^{2} + (7 + 2 i) (7 - 2 i)$

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

$6 + 27 i - 10 i - 45 \cdot - 1 + (7 + 2 i) (7 - 2 i)$

Multiply $- 45$ by $- 1$ .

$6 + 27 i - 10 i + 45 + (7 + 2 i) (7 - 2 i)$

Add $6$ and $45$ .

$51 + 27 i - 10 i + (7 + 2 i) (7 - 2 i)$

Subtract $10 i$ from $27 i$ .

$51 + 17 i + (7 + 2 i) (7 - 2 i)$

Expand $(7 + 2 i) (7 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$51 + 17 i + 7 (7 - 2 i) + 2 i (7 - 2 i)$

Apply the distributive property.

$51 + 17 i + 7 \cdot 7 + 7 (- 2 i) + 2 i (7 - 2 i)$

Apply the distributive property.

$51 + 17 i + 7 \cdot 7 + 7 (- 2 i) + 2 i \cdot 7 + 2 i (- 2 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$51 + 17 i + 49 + 7 (- 2 i) + 2 i \cdot 7 + 2 i (- 2 i)$

Multiply $- 2$ by $7$ .

$51 + 17 i + 49 - 14 i + 2 i \cdot 7 + 2 i (- 2 i)$

Multiply $7$ by $2$ .

$51 + 17 i + 49 - 14 i + 14 i + 2 i (- 2 i)$

Multiply $2 i (- 2 i)$ .

Multiply $- 2$ by $2$ .

$51 + 17 i + 49 - 14 i + 14 i - 4 i i$

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

$51 + 17 i + 49 - 14 i + 14 i - 4 (i^{1} i)$

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

$51 + 17 i + 49 - 14 i + 14 i - 4 (i^{1} i^{1})$

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

$51 + 17 i + 49 - 14 i + 14 i - 4 i^{1 + 1}$

Add $1$ and $1$ .

$51 + 17 i + 49 - 14 i + 14 i - 4 i^{2}$

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

$51 + 17 i + 49 - 14 i + 14 i - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$51 + 17 i + 49 - 14 i + 14 i + 4$

Add $49$ and $4$ .

$51 + 17 i + 53 - 14 i + 14 i$

Add $- 14 i$ and $14 i$ .

$51 + 17 i + 53 + 0$

Add $53$ and $0$ .

$51 + 17 i + 53$

Step 2

Add $51$ and $53$ .

$104 + 17 i$

Simplify (3-5i)(2+9i)+(7+2i)(7-2i)

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