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Chapter 2: Problem 57

Write the quotient in standard form. $$\frac{9-4 i}{i}$$

Short Answer

Expert verified
The simplified quotient in standard form is \(4+9i\).

Step by step solution

01

Identify the conjugate of the denominator

The denominator is \(i\). Its conjugate would simply be \(-i\). This is because 'i' is purely imaginary and doesn't have a real part, making the conjugate just its opposite.
02

Multiply with the conjugate

Multiply the numerator and the denominator by the conjugate of the denominator. So, we have: \[ \frac{(9-4i) \cdot -i}{i \cdot -i} \]. This simplifies to \[ \frac{-9i+4i^2}{-1}\]. Remember, \(i^2 = -1\), we can substitute this in the simplification.
03

Simplify and write in standard form

Substitute \(i^2\) with \(-1\) to get: \[ \frac{-9i+4(-1)}{-1}\] which is simplified to \[ \frac{-9i-4}{-1}\]. When we divide by \(-1\), we obtain \(9i+4\). In standard form, this is \(4+9i\). Hence, the quotient is \(4+9i\).

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