Find the following quotient and express the answer in standard form of a complex number.

[tex]\dfrac{5-7i}{4-3i}=\dfrac{(5-7i)(4+3i)}{(4-3i)(4+3i)}\\\\=\dfrac{5(4+3i)-7i(4+3i)}{4^2-(3i)^2}\\\\=\dfrac{20+15i-28i-21i^2}{16+9}\\\\=\dfrac{20+21-13i}{25}\\\\\large \boxed{=\dfrac{41}{25}-\dfrac{13}{25}i\\}[/tex]

Thank you.

MrRoyal MrRoyal · Answer 2 · 2021-11-04T10:03:19+01:00

Complex numbers are numbers with real and imaginary parts.

The expression in standard form of a complex number is: [tex]\mathbf{\frac{41}{25}-\frac{13}{25}i}[/tex]

The expression is given as:

[tex]\mathbf{\frac{5 - 7i}{4 - 3i}}[/tex]

Rationalize

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{5 - 7i}{4 - 3i} \times \frac{4 + 3i}{4 + 3i}}[/tex]

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{20 +15i -28i - 21i.i}{(4)^2 - (3i)^2}}[/tex]

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{20 +15i -28i - 21(-1)}{16 - 9(-1)}}[/tex]

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{20 +15i -28i + 21}{16 + 9}}[/tex]

Collect like terms

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{20 + 21+15i -28i }{16 + 9}}[/tex]

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{41-13i }{25}}[/tex]

Split

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{41}{25}-\frac{13i }{25}}[/tex]

[tex]\mathbf{\frac{5 - 7i}{4 - 3i} = \frac{41}{25}-\frac{13}{25}i}[/tex]

Hence, the expression in standard form of a complex number is: [tex]\mathbf{\frac{41}{25}-\frac{13}{25}i}[/tex]