Math  /  Numbers & Operations

Question10. What is the result of (3+2i)(1+2i)(3+2 i)-(1+2 i) ? a. -2 b. 2+0i2+0 i c. 2i2 i d. 22i2-2 i

Studdy Solution

STEP 1

1. The expression (3+2i)(1+2i)(3+2i)-(1+2i) involves basic arithmetic operations on complex numbers.
2. A complex number is in the form a+bia+bi, where aa and bb are real numbers, and ii is the imaginary unit with i2=1i^2 = -1.
3. The subtraction of complex numbers follows the rule (a+bi)(c+di)=(ac)+(bd)i(a+bi)-(c+di) = (a-c) + (b-d)i.

STEP 2

1. Separate the real and imaginary parts of the complex numbers (3+2i)(3+2i) and (1+2i)(1+2i).
2. Subtract the real parts.
3. Subtract the imaginary parts.
4. Combine the results to get the final expression in the form a+bia+bi.

STEP 3

Separate the real and imaginary parts of the complex numbers (3+2i)(3+2i) and (1+2i)(1+2i).
(3+2i) has real part 3 and imaginary part 2i (3+2i) \text{ has real part } 3 \text{ and imaginary part } 2i (1+2i) has real part 1 and imaginary part 2i (1+2i) \text{ has real part } 1 \text{ and imaginary part } 2i

STEP 4

Subtract the real parts.
31=2 3 - 1 = 2

STEP 5

Subtract the imaginary parts.
2i2i=0i 2i - 2i = 0i

STEP 6

Combine the results to get the final expression in the form a+bia+bi.
(3+2i)(1+2i)=2+0i (3+2i)-(1+2i) = 2 + 0i
Solution: The result of (3+2i)(1+2i)(3+2i)-(1+2i) is 2+0i2 + 0i, which corresponds to option (b).

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