Complex Numbers

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Step 1: Powers of i repeat every 4 terms: i, -1, -i, 1. Step 2: Divide 25 by 4. 25 = 4 × 6 + 1, so the remainder is 1. Step 3: Remainder 1 gives i. Answer: i…

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Step 1: Use the reciprocal rule for negative powers. i^-101 = 1 / i^101 Step 2: Reduce 101 by the 4-cycle. 101 = 4 × 25 + 1, so i^101 = i. Step 3: Therefore, i^-101 = 1 / i = -i Answer: -i…

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Step 1: Write each negative square root using i. sqrt(-100) = 10i sqrt(-9) = 3i Step 2: Add the results. 10i + 3i = 13i Answer: 13i…

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Step 1: Convert each square root. sqrt(-36) = 6i sqrt(-49) = 7i Step 2: Multiply. (6i)(7i) = 42i^2 Step 3: Use i^2 = -1. 42i^2 = -42 Answer: -42…

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Step 1: Convert both roots. sqrt(-3) = i sqrt(3) sqrt(-5) = i sqrt(5) Step 2: Multiply. (i sqrt(3))(i sqrt(5)) = i^2 sqrt(15) Step 3: Use i^2 = -1. i^2 sqrt(15) = -sqrt(15) Answer: -sqrt(15)…

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Use it only when at least one of a and b is non-negative. Example: sqrt(4 × 9) = sqrt(4) × sqrt(9) = 2 × 3 = 6. It should not be used blindly for negative numbers under the root. Example check: sqrt(-…

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Use the 4-cycle of powers of i. i^0 = 1 If n > 0, divide n by 4 and use the remainder: remainder 0 → 1 remainder 1 → i remainder 2 → -1 remainder 3 → -i If n < 0, first convert it to a reciprocal and …

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Step 1: Simplify the powers of i. i^3 = -i and i^2 = -1 So, 3i^3 - 2ai^2 + (1 - a)i + 5 = 3(-i) - 2a(-1) + (1 - a)i + 5 = -3i + 2a + (1 - a)i + 5 = (2a + 5) + (-2 - a)i Step 2: For the number to be re…