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: Idea:
Since $w^3 = 1$, the powers of $w$ repeat in a cycle of length 3: $w^1 = w,\ w^2,\ w^3 = 1,\ w^4 = w,\dots$.

Step 2: Find remainders of the exponents on division by 3:
10 leaves remainder 1, so $w^{10} = w$. 7 leaves remainder 1, so $w^7 = w$. 5 leaves remainder 2, so $w^5 = w^2$.

Step 3: Add up:
$(w - w) + (w^2 - w^2) + 1 = 0 + 0 + 1 = 1$.

Final Answer:
The expression equals 1, option (C). \[ \boxed{1} \]