The value of ((-1+i √(3))^15)/((1-i)^20)+((-1-i √(3))^15)/((1+i)^20) is (a) 32 (b) 64 (c) -64 (d

The value of ((-1+i √(3))^15)/((1-i)^20)+((-1-i...

Key Concepts

Polar Representation of Complex Numbers

Complex numbers can be expressed in the polar form, which represents a complex number as r*(cos(?) + i*sin(?)) or equivalently r*exp(i?). This form makes multiplication, division, and exponentiation of complex numbers much easier by allowing operations on the modulus and rotations in the complex plane through the argument.

Euler's Formula

Euler's Formula states that exp(i?) = cos(?) + i*sin(?), establishing a deep relationship between trigonometric functions and the exponential function. This formula is particularly powerful in simplifying complex expressions and is a fundamental tool in handling complex exponentiation and phasor analysis.

De Moivre's Theorem

De Moivre's Theorem provides a formula to raise complex numbers in polar form to a power: (r*exp(i?))^n = r^n * exp(i*n?). This theorem is essential for simplifying expressions involving powers and roots of complex numbers, as it reduces the problem to manipulating the modulus and the argument.

Properties of Complex Conjugates

Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. Their properties, such as having the same modulus and their product being a real number, are useful in simplifying complex expressions, particularly when expressions involve both a complex number and its conjugate.

Transcript

00:01 In the given question we have to find the value of an expression that is given as minus 1 plus square root of i square root of 3 raised to the power 15 divided by 1 minus i raised to the power 20 plus minus 1 minus i root 3 raised to the power 15 divided by 1 minus i root 3 raised to the power 15 divided by 1 plus i raised to the power 20.

00:35 So what we can do first is to take this expression that is minus 1 plus i root 3 and write it as 2 times minus 1 by 2 plus i root 3 by 2 right and now we should know that cost of the angle cause of the angle 2 pi by 3, 2 pi by 3 is equal to minus 1 by 2 and sine of 2 pi by 3 is equal to root 3 by 2.

01:12 And let's make the substitution over here.

01:15 So we will have cause of 2 pi by 3 plus i sine 2 pi by 3.

01:24 Right? so this is what we get.

01:28 And now what we can do is to take the demover's theorem, the demover's theorem by which we can write that if we have the expression, we have an expression such that cost of theta plus i sine theta is raised to a power, say n, then it is, it can be as cost of n theta plus i sign of n theta.

02:04 So let's use this property over here.

02:07 So in the above question we have the expression minus 1 plus i root 3 is raised to the power 15.

02:20 Right so minus 1 plus i root 3 is raised to the power 15 so we made the substitution of minus 1 by root 3 as cos 2 pi by 3 cost of 2 pi by 3 plus i sine 2 pi by 3 so this would be raised to 15 and now we have 2 times 2 raised to the power 15 times cost of 15 times 2 pi by 3 is 30 pi by 3 plus i sign 30 pi by 3 right so this is what we get and we need to know another theorem which is the euler's theorem by which we can write cost theta plus i sine theta as the power of e that is e raised to i theta so we have this property as well so now we can further simplify this expression and write it as 2 to the power 15 times e raised to the power 30 pi by 3 which is 30 by 3 is 10 so 10 pi right so this is the first part of the given problem next what we can do is to take the denominator of the first term that is 1 minus i raised to 20 and then we can simplify it in the same form by taking root 2 by writing it as root 2 times 1 by root 2 minus i 1 by root 2 and we know that cost of pi by 4 is equal to 1 by root 2 and sine of pi by 4 is also equal to 1 by root 2 right so now we can write this as square root of cos pi by 4 minus i sine pi by 4 and from this we can write it as cost of root 2 times cost of minus pi by 4 plus i sign of minus pi by 4 and now we can write it according to the euler's theorem as root 2 times e to the power i times minus pi by 4.

05:19 So the first term of the given question would then be written as the numerator is 2 raised to the power 15 divided 15 times e raised to the power 10 pi and over here we have the denominator which is raised to the power 20 so 1 minus i raised to the power 20 would be root 2 raised to the power 20 times e raised to i minus pi times 20 which is equal to e raised 2 raised to 10 times e raised to the power minus 5 pi i right so over here we have i over here it is 10 pi i and over here we have minus 5 pi i so the first term of the expression given in the question would be written as 2 raised to the power 15 times e raised to the power 10 pi i divided by 2 raised to the power 10 times e raised to the power minus 5 pi i.

06:43 So this is the first term right and we can similarly write the second term of the expression as we can just look at the second term of the expression as we can just look at the second term of the expression.

06:57 And it is of the form minus minus 1 minus i root 3 and when we write it just like the first form it would be it would be like this right it would be in the form 2 raised 2 raised to the power 15 times e raised to the power minus 10 pi since we have minus 1 minus i root 3 and it would be divided by root 2 times e raised to the power 5 pi i since it is 1 plus i right so this is the difference that would come in the second term so now that we have simplified both terms of the given question let's add them so let's add these terms...

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