Evaluate \([i^{18}+(\frac{1}{i})^{25}]^3.\)
Solution and Explanation
\([i^{18}+(\frac{1}{i})^{25}]^3\)
=\([i^{4×4+2}+(\frac{1}{i^{4×4+1}})]^3\)
=\([i^{(4)^4.i^2}+\frac{1}{(i^{4})^6.i}]^3\)
\(=[i^2+\frac{1}{i}]^3\) \([i^{-4}=1]\)
\(=[-1+\frac{1}{i}×\frac{1}{i}]^3\) \([i^4=-1]\)
\(=[-1+\frac{i}{i^2}]^3\)
\(=[-1-i]^3\)
=\((-1)^3[1+i]^3\)
\(=[1^3+i^3+3.1.i(1+i)]\)
\(=-[1+i^3+3i+3i^2]\)
\(=-[1+i^3+3i+3i^2]\)
\(=-[1-i+3i-3]\)
\(=-[-2+2i]\)
\(=2-2i\)
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.