\(\text{Find the number of non-zero integral solutions of the equation }|1-i|^x=2^2.\)
Solution and Explanation
\(|1-i|^x=2^2\)
\(⇒(\sqrt1^2+(-1)^2)=2^x\)
\(⇒(\sqrt2)^x=2^2\)
\(⇒2^{\frac{x}{2}}=2^x\)
\(⇒\frac{x}{2}=x\)
\(⇒x=2x\)
\(⇒2x-x=0\)
\(⇒x=0\)
\(\text{Thus, 0 is the only integral solution of the given equation. Therefore, the number of non-zero integral solutions of the given equation is 0.}\)
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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.