Question

Match each power of i with its multiplicative inverse.

1) i
2) i^2
3) i^3
4) i^4

Each number has one of the following answers answers cannot be used more than once.
Answer 1) i
Answer 2) 1
Answer 3) -i
Answer 4) -1

Answer 1

By definition we have to:
 In mathematics, the inverse multiplicative, reciprocal or inverse of a non-zero number x, is the number, denoted as 1/x or x -1, which multiplied by x gives 1.
 We have then:

 1) i
 The multiplicative inverse is:
 $\frac{1}{i}$
 Rewriting we have:
 $\frac{1}{i} \frac{i}{i}$
 $\frac{i}{i^2}$
 $\frac{i}{-1}$
 $-i$
 Answer 3) -i

 2) i^2
 The multiplicative inverse is:
 $\frac{1}{i^2}$
 Rewriting we have:
 $\frac{1}{-1}$
 $-1$
 Answer 4) -1

 3) i^3
 The multiplicative inverse is:
 $\frac{1}{i^3}$
 Rewriting we have:
 $\frac{1}{i*i^2}$
 $\frac{1}{i^2}*\frac{1}{i}$
 $\frac{1}{-1}*\frac{1}{i}*\frac{i}{i}$
 $(-1)*\frac{i}{i^2}$
 $(-1)* \frac{i}{-1}$
 $i$
 Answer 1) i

 4) i^4
 The multiplicative inverse is:
 $\frac{1}{i^4}$
 Rewriting we have:
 $\frac{1}{i^2i^2}$
 $\frac{1}{(-1)(-1)}$
 $\frac{1}{1}$
 $1$
 Answer 2) 1

Answer 2

Answer:

1. $i=\frac{1}{i}*\frac{-i}{-1}=-i$
2. $i^2=\frac{1}{i^2}*\frac{-i^2}{-i^2}=-1$
3. $i^3=\frac{1}{i^3}*\frac{-i^3}{-i^3}=i$
4. $i^4=\frac{1}{i^4}*\frac{-i^4}{-i^4}=1$

Step-by-step explanation:

To find the multiplicative inverse of a complex number, you do as follows:

The inverse is found by reciprocating the original complex number. ($\frac{1}{complex number}$) Multiply the numerator and denominator of the reciprocal by conjugate of the denominator and simplify.

Hope this helps!!