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Chapter 2: Problem 56

Write the quotient in standard form. $$\frac{6-7 i}{1-2 i}$$

Short Answer

Expert verified
The quotient of \(\frac{6-7 i}{1-2 i}\) in standard form is \(4+i\)

Step by step solution

01

Identify the Conjugate

In any complex number a+bi, the conjugate is given by a-bi. So, the conjugate of \(1-2i\) is \(1+2i\).
02

Multiply the numerator and the denominator by the conjugate

Multiply the numerator \(6-7i\) and the denominator \(1-2i\) both by the conjugate \(1+2i\). This gives \((6-7i)(1+2i)\) over \((1-2i)(1+2i)\).
03

Expand the numerator and the denominator

Expand to remove the brackets in the numerator and the denominator. This results in \(6+12i-7i-14i^2\) over \(1-2i+2i-4i^2\).
04

Simplify the numerator and the denominator

Simplify both the numerator and denominator remembering that \(i^2\) equals -1. We have \(6+5i+14\) over \(1+4\), which simplifies further to \(20+5i\) over \(5\).
05

Divide the numerator by the denominator

Finally, divide the numerator by the denominator. The result in this case would be \(4+i\) .

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