The solutions of x4=1 (x to the power 4 equals 1) are given by x= 1, -1, i, and -i where i=√-1 is the imaginary complex number. In this post, we will learn how to find the roots of x4=1.
Solutions of x4=1
To solve the given equation x4=1, we need to follow the below steps given here.
Step 1: First, write x4=1 in the form of x4-1=0, then apply the formula of a2-b2.
⇒ x4 -1 =0
⇒ (x2)2 -12 =0
⇒ (x2 -1)(x2+1) =0 [applying the formula of a2-b2=(a-b)(a+b) ]
So either x2-1=0 or x2+1=0 …(∗)
Step 2: Now, we will solve the above two equations.
First, solve x2-1=0
⇒ (x-1) (x+1)=0
So either x-1=0 or x+1=0
∴ x= 1, -1 …(∗)
Step 3: Now, solve x2+1=0
⇒ x2= -1
⇒ x2= i2
Taking square roots, x= i, -i …(∗∗)
Therefore, from (∗) and (∗∗), we conclude that the solutions of x4=1 are -1, 1, i, and -i where i = √-1.
So there are only two real roots of x^4=1. Ans there are two complex roots of x^4=1.
Roots of x4=1
Note that the solutions of the equation x4=1 are the same as the roots of x4=1. Thus, from above we deduce that
the roots of x3=1 are ±1 and ±i
where i = √-1 is the imaginary complex number.
Q1: What are the solutions of x4=1?
Answer: The solutions of x^4=1 are x=±1 and ±i where i = √-1 is an imaginary complex number.
Q2: What are the roots of x4=1?
Answer: The roots of x^4=1 are 1, -1, i, and -i where i=√-1 is the imaginary complex number.
Q3: Find the number of real roots of x4=1.
Answer: The number of real roots of x^4=1 is two.
Q4: Find the number of complex roots of x4=1.
Answer: The number of complex roots of x^4=1 is two.