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Chapter 2: Problem 60

Write the quotient in standard form. $$\frac{5 i}{(2+3 i)^{2}}$$

Short Answer

Expert verified
The quotient in standard form is \(\frac{-30}{61}i + \frac{5}{61}\).

Step by step solution

01

Squaring the Denominator

Firstly, square the complex number in the denominator to simplify it. This means multiplying \(2+3i\) by itself: \( (2+3i) \cdot (2+3i) \). When multiplying complex numbers, treat \(i\) element as you would \(x\) in a normal binomial, but remember that \(i^2\) is -1. Thus, \( (2+3i)^2 = 4 + 2(2)(3)i + (3i)^2 = 4 + 6i - 9 \). This simplifies to \( -5 + 6i \).
02

Dividing the complex numbers

Now we can divide our numerator by the simplified denominator: \( \frac{5i}{-5+6i} \). In order to simplify the fraction further, we multiply both the numerator and the denominator by the conjugate of the denominator, which changes the sign of the \(i\) term: \( \frac{5i \cdot (-5-6i)}{-5+6i \cdot (-5-6i)} \).
03

Simplifying the fraction

When we perform the operations in the numerator and the denominator, we get the following: \( \frac{-30i - 5i^2}{25 - 36i^2} = \frac{-30i + 5}{61} \). The term \(5i^2\) changes to 5 because \(i^2\) is -1, and the term \(-36i^2\) changes to 36, for the same reason.
04

Final quotient in standard form

Finally, we write the quotient in standard form: \( \frac{-30}{61}i + \frac{5}{61} \).

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