Question

If \(w\) is a complex cube root of unity , then the value of \(w^{10}-w^7+w^5-w^2+1\) is.

• A. \(0\)
• B. \(-1\)
• C. \(1\)
• D. \(w\)

Hint:

Use w^3 = 1 to reduce every power of w to w or w squared.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A complex cube root of unity \(w\) satisfies \(w^3 = 1\) and \(1 + w + w^2 = 0\). Any power of \(w\) can be reduced by removing multiples of 3 from the exponent.

Step 2: Reduce each power:
\(w^{10} = w^{9} \cdot w = w\).
\(w^{7} = w^{6} \cdot w = w\).
\(w^{5} = w^{3} \cdot w^2 = w^2\).
\(w^2\) stays as it is.

Step 3: Substitute:
\[ w^{10} - w^7 + w^5 - w^2 + 1 = w - w + w^2 - w^2 + 1 = 1 \]

Step 4: Why the other options are wrong.
The value 0 would result if the constant 1 were missing. The value -1 and \(w\) come from sign or exponent slips, such as treating \(w^{10}\) as \(w^{2}\) or forgetting that the terms cancel in pairs.

Final Answer:
The value of the expression is \(1\), option (C). \[ \boxed{1} \]