Math Problem Statement

The value of i^146 is equal to ______.

Solution

The question asks for the value of i146i^{146}, where ii is the imaginary unit, defined as i=1i = \sqrt{-1}.

To solve this, we use the cyclical property of powers of ii: i1=i,i2=1,i3=i,i4=1,and then it repeats every 4 powers.i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and then it repeats every 4 powers.}

To find i146i^{146}, we calculate the remainder of 146 when divided by 4: 146÷4=36 remainder 2146 \div 4 = 36 \text{ remainder } 2 Thus, i146=i2i^{146} = i^2.

From the cyclical pattern, we know: i2=1i^2 = -1

Therefore, the value of i146i^{146} is 1\boxed{-1}, which corresponds to option (B).

Do you want more details or have any questions? Here are five related questions to deepen your understanding:

  1. How do powers of ii form a cycle?
  2. What is the value of i202i^{202}?
  3. How is the remainder important when dealing with powers of ii?
  4. Can you explain the properties of imaginary numbers?
  5. How do complex numbers apply to real-world problems?

Tip: For higher powers of ii, always reduce the exponent modulo 4 to use the repeating pattern.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Powers of Imaginary Unit

Formulas

i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1

Theorems

Cyclic Nature of Powers of i

Suitable Grade Level

Grades 9-12