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Chapter 2: Problem 127

\(w \neq 1\) is cube root of 1 and \(\left(1+w^{2}\right)^{n}=\left(1+w^{4}\right)^{n}\) then the smallest positive value of \(\mathrm{n}\) is (a) 2 (b) 3 (c) 5 (d) 6

Short Answer

Expert verified
The smallest positive value of \(n\) that satisfies the equation is \(n = 1\), which is not included in the given options.

Step by step solution

01

Find the possible values of w

Since \(w\) is a cube root of 1, and \(w \neq 1\), we know that \(w^3 = 1\). The three cube roots of 1 are: 1. \(1 (\text{which should not be considered as given } w \neq 1)\) 2. \(w = e^{\frac{2\pi i}{3}}\) 3. \(w = e^{\frac{4\pi i}{3}}\) Since we can't consider \(w=1\), let's replace \(w\) by \(e^{\frac{2\pi i}{3}}\) or \(e^{\frac{4\pi i}{3}}\) in the given equation: \(\left(1+w^{2}\right)^{n}=\left(1+w^{4}\right)^{n}\)
02

Simplify the equation using \(w = e^{\frac{2\pi i}{3}}\)

Replacing \(w\) with \(e^{\frac{2\pi i}{3}}\) in the given equation, we have: \(\left(1+\left(e^{\frac{2\pi i}{3}}\right)^2\right)^{n}=\left(1+\left(e^{\frac{2\pi i}{3}}\right)^4\right)^{n}\) The next step is to simplify the exponents and the equation: \[\left(1+e^{\frac{4\pi i}{3}}\right)^{n}=\left(1+e^{\frac{8\pi i}{3}}\right)^{n}\] Using the fact that both sides of the equation must be equal, we can deduce that this equation either holds true for this value of \(w\) or doesn't. If it holds true, then the equality must hold for any value of \(n\), and consequently, the smallest positive value of \(n\) would be 1. If it doesn't, we move on to the next possible value of \(w\).
03

Simplify the equation using \(w = e^{\frac{4\pi i}{3}}\)

Replacing \(w\) with \(e^{\frac{4\pi i}{3}}\) in the given equation, we have: \(\left(1+\left(e^{\frac{4\pi i}{3}}\right)^2\right)^{n}=\left(1+\left(e^{\frac{4\pi i}{3}}\right)^4\right)^{n}\) Now, let's simplify the exponents and the equation: \[\left(1+e^{\frac{8\pi i}{3}}\right)^{n}=\left(1+e^{\frac{16\pi i}{3}}\right)^{n}\]
04

Examine the simplified equation

If we examine the two values of \(w\) and corresponding equations we obtained in Step 2 and Step 3, we will notice that in either case, the equation equals to itself for any value of \(n\). This means, for each \(w\), the given condition is satisfied. However, since we are asked to find the smallest positive value of \(n\), we should choose \(n = 1\) as the smallest positive value that satisfies the equation. The answer is: \(n = 1\), which is not in the given options. There could be an error in the question or the options provided. Make sure to verify the question and the given options.

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