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Simplify the expression: ت + ت2 + ت3 + ت4

Mathematics

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Here's the solution for question 21:

  1. Identify the powers of the imaginary unit ii (represented by 'ت' in Arabic).

    We know that:
    i1=ii^1 = i
    i2=−1i^2 = -1
    i3=i2⋅i=−1⋅i=−ii^3 = i^2 \cdot i = -1 \cdot i = -i
    i4=i2⋅i2=(−1)⋅(−1)=1i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1

  2. Substitute these values into the numerator of the given expression.

    The numerator is ت+ت2+ت3+ت4\text{ت} + \text{ت}^2 + \text{ت}^3 + \text{ت}^4.
    Substituting the values:
    Numerator $=

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