Question

Let $z = 2 e^{10 \pi i/21} $ and $w = e^{\pi i/7}$. Then if \[(z-w)^6 = r e^{i\theta},\]where $r \geq 0$ and $0 \leq \theta < 2\pi$, what is the ordered pair $(r, \theta)$?

Answer 1

We are dealing with Complex Numbers in Euler Form.

I am putting the question here to make it more readable:

$Let\ z = 2 e^{10 \pi i/21} and w = e^{\pi i/7} . Then if \[(z-w)^6 = r e^{i\theta},\]where r \geq 0 and 0 \leq \theta < 2\pi , what is the ordered pair (r, \theta) ?$

So we have:

$z=2e^{\frac{10\pi i}{21}} \\ \\ \\ w=e^{\frac{\pi i}{7}}$

We need to compute $(z-w)^6$ in Euler Form. Let us first see what $(z-w)$ simplifies to.

And so:

$z-w\\\\\\=2e^{\frac{10\pi i}{21}}-e^{\frac{\pi i}{7}}\\\\\\=e^{\frac{\pi i}{7}}(2e^{\frac{\pi i}{3}}-1)$

Writing the term inside brackets in polar form:

$=e^{\frac{\pi i}{7}}(2\left(\cos \frac{\pi}{3}+i\sin\frac{\pi}{3}\right)-1)\\\\\\=e^{\frac{\pi i}{7}}(2\left(\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)-1)\\\\\\=e^{\frac{\pi i}{7}}(1+i\sqrt{3}-1)\\\\\\\implies (z-w)=\sqrt{3}ie^{\frac{\pi i}{7}}$

Raising it to the power 6, we have:

$(z-w)^6\\\\\\=\left(\sqrt{3}ie^{\frac{\pi}{7}}\right)^6\\\\\\=27i^6e^{\frac{6\pi i}{7}}\\\\\\=-27e^{\frac{6\pi i}{7}} \quad \mathsf{[\ i^6=i^2.i^4=-1.1=-1\ ]}\\\\\\\textsf{We can write \sf -1 as \sf e^{\pi i} }\\\\\\=27e^{\frac{6\pi i}{7}}e^{\pi i}\\\\\\=27e^{\frac{6\pi i}{7}+\pi i}\\\\\\=27e^{\frac{13\pi i}{7}}\\\\\\\\\implies \Large \textsf{ (z-w)^6=27e^{\frac{13 \pi i}{7}} }$

Comparing with the standard form $re^{i\theta}$, we get:

$\Large \textsf{r=27} \\\\\\ \Large \textsf{ \theta=\dfrac{13 \pi}{7} }$

Hence, we have the answer:

$\Huge \boxed{\sf (r,\theta)=\left(27,\frac{13\pi}{7}\right)}$

Answer 2

Answer:

QUESTION:

What is the mass number of an atom with 6 protons and 8 neutrons?

Answer:

$\huge{ \boxed{ \bold{ \sf{14 \: amu}}}}$

Explanation:

✑ First, Let's explore about ATOMIC MASS :

↱ An atom consists of electrons , protons and neutrons. The mass of electron is extremely small and it is taken as a negligible mass compared to the mass of a proton or neutron. Thus , The atomic mass is defined as the sum of the number of protons and neutrons present in the nucleus of an atom of that element. If is also called atomic weight. It is denoted by A. The nucleus determines the mass of an atom, while the number of electrons determines the size of the atom of the element.

Mathematically,

Atomic mass = Number of protons + Number of neutrons

i.e $\boxed{ \sf{A = p^{ + } + n^{0}}}$

———————————————————————

✎ Now , Let's find the atomic mass of atom having 6 protons and 8 neutrons :

☞ Atomic mass = No. of protons + No. of neutrons

➵ Atomic mass = 6 + 8

➵ Atomic mass =$\boxed{ \sf{14 \: amu}}$

And we're done !

Hope I helped!

Have a wonderful day! ツ

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