QuestionAugust 25, 2025

4.) Simplify completely: 3i^287+2i^943=

4.) Simplify completely: 3i^287+2i^943=
4.) Simplify completely: 3i^287+2i^943=

Solution
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Answer

-5i Explanation 1. Simplify powers of i The powers of i cycle every 4: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. Therefore, find the remainder when dividing the exponents by 4. 2. Calculate i^{287} 287 \mod 4 = 3, so i^{287} = i^3 = -i. 3. Calculate i^{943} 943 \mod 4 = 3, so i^{943} = i^3 = -i. 4. Combine terms Substitute back: 3(-i) + 2(-i) = -3i - 2i = -5i.

Explanation

1. Simplify powers of $i$<br /> The powers of $i$ cycle every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Therefore, find the remainder when dividing the exponents by 4.<br />2. Calculate $i^{287}$<br /> $287 \mod 4 = 3$, so $i^{287} = i^3 = -i$.<br />3. Calculate $i^{943}$<br /> $943 \mod 4 = 3$, so $i^{943} = i^3 = -i$.<br />4. Combine terms<br /> Substitute back: $3(-i) + 2(-i) = -3i - 2i = -5i$.
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