Question

consider the complex number z=5-√3i÷4+2√3i​

Answer 1

Complex numbers

The given complex number is

$\quad\quad z=\frac{5-\sqrt{3}i}{4+2\sqrt{3}i}$

We have to write $z$ in the form $a+ib.$

Step-by-step explanation:

The given complex number is

• $z=\frac{5-\sqrt{3}i}{4+2\sqrt{3}i}$

Multiply the denominator by the conjugate complex number $(4-2\sqrt{3}i).$

• $z=\frac{(5-\sqrt{3}i)(4-2\sqrt{3}i)}{(4+2\sqrt{3}i)(4-2\sqrt{3}i)}$

Use the identity: $(a+b)(a-b)=a^{2}-b^{2}$ in denominator and multiply the terms of the numerator.

• $z=\frac{20-10\sqrt{3}i-4\sqrt{3}i+6i^{2}}{16-12i^{2}}$

We must know that: $i^{2}=-1.$

• $z=\frac{20-16\sqrt{3}i-6}{16+12}$

• $z=\frac{14-16\sqrt{3}i}{28}$

• $z=\frac{14}{28}-\frac{16\sqrt{3}}{28}i$

• $z=\frac{1}{2}-\frac{4\sqrt{3}}{7}i$

This is in the form $a+ib.$

Answer 2

Complex numbers

The given complex number is

${\quad\quad z=\frac{5-\sqrt{3}i}{4+2\sqrt{3}i}z= }$

${4+2^{3}i5−3i}$

We have to write zz in the form a+ib.a+ib.

Step-by-step explanation:

The given complex number is

${z=\frac{5-\sqrt{3}i}{4+2\sqrt{3}i}z= }$

${ 4+2^{3}i5−3i }$

Multiply the denominator by the conjugate complex number

${(4-2\sqrt{3}i).(4−2^{3}i)}$

${z=\frac{(5-\sqrt{3}i)(4-2\sqrt{3}i)}{(4+2\sqrt{3}i)(4-2\sqrt{3}i)}z=}$

${(4+2^{3}i)(4−2^{3}i)(5−3i)(4−2^{3}i)}$

Use the identity:

${(a+b)(a-b)=a^{2}-b^{2}(a+b)(a−b)=a^{2}−b^{2}}$

In denominator and multiply the terms of the numerator.

${z=\frac{20-10\sqrt{3}i-4\sqrt{3}i+6i^{2}}{16-12i^{2}}z= }$

${16−12i^{2}20−10^{3}i−4^{3}i+6i^{2}}$

We must know that:

${i^{2}=-1.i^{2}=−1}$

${z=\frac{20-16\sqrt{3}i-6}{16+12}z=}$

${16+12+20−16^{3}i−6}$

${z=\frac{14-16\sqrt{3}i}{28}z=}$

${28×14−16^{3}i}$

${z=\frac{14}{28}-\frac{16\sqrt{3}}{28}iz= }$

${28+14−28×16^{3}i}$

${z=\frac{1}{2}-\frac{4\sqrt{3}}{7}iz= }$

${21−74^{3}i}$

$\bf{ This \: is \: in \: the \: form \: a+ib.a+ib. }$