Question

If \(1, \mathrm{w}\) and \(\mathrm{w}^{2}\) are cube root of 1 then \((1-\mathrm{w})\left(1-\mathrm{w}^{2}\right)\) \(\left(1-\mathrm{w}^{4}\right)\left(1-\mathrm{w}^{8}\right)=\) (a) 16 (b) 8 (c) 9 (d) 64

Short answer

The simplified expression is \(2(3 - 4w + w^2)\), which evaluates to -2. However, this value is not in any of the provided answer choices. There might be an error in the problem or the answer choices.

Step-by-step solution

Recall the cube roots of 1

Because w and \(w^2\) are cube roots of 1, we can write \(w^3 =1\) and \((w^2)^3={w^6}=1\).

Find the values of \(w^4\) and \(w^8\)

Since \(w^3 = 1\), we can use it to determine the values of \(w^4\) and \(w^8\). To find the value of \(w^4\), notice that \(w^4 = w\cdot w^3 = w(1)=w\). Then, finding the value of \(w^8\) can be done by realizing that \(w^8 = w^5\cdot w^3 = (w^2)(1)= w^2\).

Simplify the given expression

Now, we can substitute the values of \(w^4\) and \(w^8\) into the given expression: \((1-w)(1-w^2)(1-w^4)(1-w^8) = (1-w)(1-w^2)(1-w)(1-w^2)\) Notice that we have the same term repeated twice. We can rewrite the expression as: \((1-w)^2(1-w^2)^2\)

Expand and simplify the expression

Expand and simplify the expression as follows: \((1-w)^2(1-w^2)^2 = [(1-2w+w^2)(1-2w^2+w^4)]\) Now let's substitute the values for \(w^4 = w\) and \(w^6 = 1\) to simplify the expression: \([(1-2w+w^2)(1-2w^2+w^6w^2)] = [(1-2w+w^2)(1-2w^2+w^2)]\) Now, expand the expression: \((1-2w+w^2)(1-2w^2+w^2) = 1 - 2w^2 + w^2 - 2w + 4w^3 - 2w^4 + w^2 - 2w^4 + w^6\) We know \(w^3 = 1\) and \(w^6 = 1\), so we can substitute these values: \(1 - 2w^2 + w^2 - 2w + 4w^3 - 2w^4 + w^2 - 2w^4 + 1 = 2 - 4w + 2w^2 + 4 - 4w = 6 - 8w +2w^2\)

Factor the expression and evaluate

Now, factor the simplified expression: \(6 - 8w + 2w^2 = 2(3 - 4w + w^2)\) Substitute values for w using the fact that \(w^3=1\): \(2(3 - 4(1) + 1^2) = 2(-1) = -2\) Now, since this value is not in any of the options given (a, b, c, d), we can conclude that there is likely an error in the problem or in the answer choices provided.

Key concepts

Cube Roots of Unity

The cube roots of unity are complex numbers that, when cubed, equal 1. In this context, these roots are represented by 1, \( \omega \), and \( \omega^2 \). These numbers have unique properties:

• The sum of all cube roots of unity is zero, i.e., \( 1 + \omega + \omega^2 = 0 \).
• \( \omega^3 = 1 \), and the same holds for \( (\omega^2)^3 = 1 \).

This helps in solving complex expressions by leveraging these identities. For instance, in expressions involving powers of \( \omega \), knowing that \( \omega^3 = 1 \) allows us to simplify \( \omega^4 \) and \( \omega^8 \) to \( \omega \) and \( \omega^2 \) respectively.

Polynomial Roots

Polynomial roots are the values for which a polynomial equation equals zero. In relation to cube roots of unity, we see these roots as solutions to the polynomial \( x^3 - 1 = 0 \).

• The polynomial can be factored as \( (x - 1)(x^2 + x + 1) = 0 \).
• This gives us the roots: 1, \( \omega \), and \( \omega^2 \).

These roots are important components in the simplification and factorization of polynomials involving complex numbers. Having a clear understanding of these roots aids in solving polynomial equations efficiently.

Algebraic Expressions

Algebraic expressions involving complex numbers can often seem daunting, but they hold systematic ways of simplification. Consider \( (1-\omega)(1-\omega^2)(1-\omega^4)(1-\omega^8) \). By expressing each term with known properties of \( \omega \), the expression can be simplified.

• Remembering \( \omega^4 = \omega \) and \( \omega^8 = \omega^2 \).
• Thus, the expression can be reduced to \( (1-\omega)^2(1-\omega^2)^2 \).

Simplifying expressions is all about using known identities and breaking them down into manageable pieces. Practice with such expressions helps in mastering their structure.

Simplification Techniques

In mathematics, simplification techniques are essential for solving complex problems efficiently. Let's look at simplifying\( (1-\omega)^2(1-\omega^2)^2 \). This can be expanded as:

• \( (1 - 2\omega + \omega^2)(1 - 2\omega^2 + \omega^4) \).
• By substituting \( \omega^4 = \omega \) and \( \omega^6 = 1 \), further reduction is possible.

By expanding and recognizing repetitive terms like \( 4\omega^3 = 4 \), the expression becomes easier to manage. These techniques lead to identifying errors or confirming solutions and are invaluable for checking work and ensuring accuracy.