An algebra problem by John Michael Gogola
Algebra
Level
3
If = , what is the value of
The answer is -1024.
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Since we know that i = − 1 , now squaring both sides we get i 2 = − 1 . Now solving the given expression further:
\(\begin{array}{} (1 - i)^{20} & = ((1-i)^2)^{10} \\ &= (-2i)^{10} \qquad \left[ \because (1 - i )^{2} = 1 - 2i + i^{2} = 1 - 2i -1 = -2i \right] \\& = 2^{10} \cdot i^{10} \\ &= -1024. \square \end{array}\)