Solved on Dec 24, 2023

What is the number of real fifth roots of 0?

STEP 1

Assumptions
1. We are looking for the number of real fifth roots of 0.
2. A real fifth root of a number x x is a real number y y such that y5=x y^5 = x .

STEP 2

Understand the definition of a fifth root.
A fifth root of a number x x is a number y y such that when y y is raised to the power of 5, the result is x x . This can be written as:
y5=xy^5 = x

STEP 3

Apply the definition to the number 0.
We want to find a real number y y such that:
y5=0y^5 = 0

STEP 4

Consider the properties of real numbers and multiplication.
For any real number y y , the only way y5 y^5 can equal 0 is if y y itself is 0. This is because 0 multiplied by any number is 0, and any non-zero number raised to any power is non-zero.

STEP 5

Conclude that 0 is the only real fifth root of 0.
Since no other real number except for 0 raised to the fifth power will result in 0, we can conclude that 0 has exactly one real fifth root, which is 0 itself.
y=0y = 0
Therefore, the number of real fifth roots that 0 has is 1.

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