The solutions of x^{4}=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 x^{4}=1.

## Solutions of x^{4}=1

To solve the given equation x^{4}=1, we need to follow the below steps given here.

**Step 1**: First, write x^{4}=1 in the form of x^{4}-1=0, then apply the formula of a^{2}-b^{2}.

x^{4}=1

⇒ x^{4 }-1 =0

⇒ (x^{2})^{2} -1^{2} =0

⇒ (x^{2 }-1)(x^{2}+1) =0 [applying the formula of a^{2}-b^{2}=(a-b)(a+b) ]

So either x^{2}-1=0 or x^{2}+1=0 **…(∗)**

**Step 2**: Now, we will solve the above two equations.

First, solve x^{2}-1=0

⇒ x^{2}-1^{2}=0

⇒ (x-1) (x+1)=0

So either x-1=0 or x+1=0

∴ x= 1, -1 …(∗)

**Step 3**: Now, solve x^{2}+1=0

⇒ x^{2}= -1

⇒ x^{2}= i^{2}

Taking square roots, x= i, -i …(∗∗)

Therefore, from** **(∗) and (∗∗), we conclude that the solutions of x^{4}=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 x^{4}=1

Note that the solutions of the equation x^{4}=1 are the same as the roots of x^{4}=1. Thus, from above we deduce that

the roots of x^{3}=1 are ±1 and ±i

where i = √-1 is the imaginary complex number.

**ALSO READ:**

How to solve quadratic equations

## FAQs

### Q1: What are the solutions of x^{4}=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 x^{4}=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 x^{4}=1.

Answer: The number of real roots of x^4=1 is two.

### Q4: Find the number of complex roots of x^{4}=1.

Answer: The number of complex roots of x^4=1 is two.