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QuestionEvaluate the quotient and express it as a+bia + bi: (7+2i)(8−i)2+i\frac{(7+2 i)(8-i)}{2+i}.

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, which are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit with the property that i^ = -1. . We are asked to simplify the given expression to the form a + bi.

STEP 2

First, we need to multiply the two complex numbers in the numerator. We can do this using the distributive property, which states that for any three numbers a, b, and c, the equation a*(b + c) = a*b + a*c holds.
(7+2i)(8−i)=7∗8+7∗(−i)+2i∗8+2i∗(−i)(7+2i)(8-i) =7*8 +7*(-i) +2i*8 +2i*(-i)

STEP 3

Now, simplify the above expression.
(7+2i)(8−i)=56−7i+16i−2i2(7+2i)(8-i) =56 -7i +16i -2i^2

STEP 4

Remember that i^2 = -1. Substitute -1 for i^2 in the equation.
(7+2i)(8−i)=56−7i+16i+2=58+9i(7+2i)(8-i) =56 -7i +16i +2 =58 +9i

STEP 5

Now, we need to divide the result by the complex number in the denominator. To do this, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi.
58+9i2+i=(58+9i)(2−i)(2+i)(2−i)\frac{58 +9i}{2 + i} = \frac{(58 +9i)(2 - i)}{(2 + i)(2 - i)}

STEP 6

Now, multiply the two complex numbers in the numerator using the distributive property.
(58+9i)(2−i)=58∗2+58∗(−i)+9i∗2+9i∗(−i)(58 +9i)(2 - i) =58*2 +58*(-i) +9i*2 +9i*(-i)

STEP 7

implify the above expression.
(58+9i)(2−i)=116−58i+18i−9i2(58 +9i)(2 - i) =116 -58i +18i -9i^2

STEP 8

Substitute -1 for i^2 in the equation.
(58+i)(2−i)=116−40i+=125−40i(58 +i)(2 - i) =116 -40i + =125 -40i

STEP 9

Now, multiply the two complex numbers in the denominator using the formula (a + b)(a - b) = a^2 - b^2.
(2+i)(2−i)=22−i2(2 + i)(2 - i) =2^2 - i^2

STEP 10

Substitute - for i^2 in the equation.
(2+i)(2−i)=4−(−)=5(2 + i)(2 - i) =4 - (-) =5

STEP 11

Now, divide the result in the numerator by the result in the denominator.
125−40i5=25−8i\frac{125 -40i}{5} =25 -8iSo, the simplified form of the given expression is25 -8i.

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