Problem 76

Question

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

Step-by-Step Solution

Verified

Answer

The statement is false.

1Step 1: Simplify each power of i

Since powers of $ i $ repeat every 4, we can simplify by taking each exponent modulo 4. The modulo operation gives the remainder when the number is divided by 4. So, $ i^{44} $ is equivalent to $ i^{0} $. Similarly for other terms, $ i^{150} = i^{2}, i^{74} = i^{2}, i^{109} = i^{1}, i^{61} = i^{1} $.

2Step 2: Substitute the values of i

Substitute the values of $ i^0, i^1, i^2 $ into the equation. $i^0 = 1, i^1 = i, i^2 = -1$. So the expression becomes $1 + (-1) - (-1) - i + i $.

3Step 3: Simplification and checking

Simplify the expression, which results in $ 1 $. This is not equal to $ -1 $, so the statement is false.

Problem 76

Problem 77