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Chapter 2: Problem 90

Simplify the complex number and write it in standard form. $$(-i)^{6}$$

Short Answer

Expert verified
The simplified form of \((-i)^6\) is \(-1\).

Step by step solution

01

Identify the base

In our problem, we can see that the base is \(-i\). However, for simplifying, we will change this form into \(i^{-1}\) to maintain our operations within the cyclical nature of \(i\). The original equation will then become \((i^{-1})^6\).
02

Simplify the expression

By using the mathematical rule, \((a^m)^n = a^{m*n}\), we can simplify the expression to \(i^{-6}\), which is equivalent to \(1/i^6\).
03

Simplify the expression using cyclical nature

Remembering our rule about the cyclical powers of \(i\) that every fourth power (starting from \(i^4\)) is 1, so \(i^4 = 1\), \(i^8 = 1\), \(i^{12} = 1\), etc. To figure out what \(i^6\) is, we divide 6 by 4 which gives us 1 (whole number). We subtract \(1*4\) from 6, which gives us a remainder of 2. This means \(i^6 = i^2\). Hence, \(1/i^6\) becomes \(1/-1 = -1\).

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