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QuestionCalculate the product of (−6−4i)(-6-4i) and its conjugate (−6+4i)(-6+4i).

Studdy Solution

STEP 1

Assumptions1. We have a complex number −6−4i-6-4i. . The conjugate of a complex number a+bia+bi is a−bia-bi.

STEP 2

First, we need to find the conjugate of the given complex number. The conjugate of −6−4i-6-4i is −6+4i-6+4i.

STEP 3

Now, we need to find the product of the complex number and its conjugate. The product of a complex number a+bia+bi and its conjugate a−bia-bi is (a+bi)(a−bi)(a+bi)(a-bi).

STEP 4

Substitute −6−4i-6-4i and −6+4i-6+4i into the formula.
(−6−4i)(−6+4i)(-6-4i)(-6+4i)

STEP 5

Expand the product using the distributive property.
(−−4i)(−+4i)=(−)(−)+(−)(4i)−(4i)(−)−(4i)(4i)(--4i)(-+4i) = (-)(-) + (-)(4i) - (4i)(-) - (4i)(4i)

STEP 6

implify the expression.
(−6−4i)(−6+4i)=36−24i+24i−16i2(-6-4i)(-6+4i) =36 -24i +24i -16i^2

STEP 7

Remember that i2=−1i^2 = -1. Substitute −1-1 for i2i^2 in the expression.
(−6−4i)(−6+4i)=36−24i+24i+16(-6-4i)(-6+4i) =36 -24i +24i +16

STEP 8

implify the expression.
(−6−4i)(−6+4i)=52(-6-4i)(-6+4i) =52
So, the product of (−6−4i)(-6-4i) and its conjugate is 5252.

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