To solve the equation [tex]\( y^{20} = y^2 \)[/tex], we need to find the values of [tex]\( y \)[/tex] that satisfy this equality.
### Step-by-Step Solution:
1. Understand the equation:
We need to solve for [tex]\( y \)[/tex] in the equation [tex]\( y^{20} = y^2 \)[/tex].
2. Rewrite the equation:
[tex]\[
y^{20} = y^2
\][/tex]
3. Bring all terms to one side of the equation:
[tex]\[
y^{20} - y^2 = 0
\][/tex]
4. Factor the expression:
Factor out [tex]\( y^2 \)[/tex] from the left-hand side:
[tex]\[
y^2 (y^{18} - 1) = 0
\][/tex]
5. Solve for y:
For the product to be zero, either [tex]\( y^2 = 0 \)[/tex] or [tex]\( y^{18} - 1 = 0 \)[/tex].
- Case 1:
[tex]\[
y^2 = 0
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
y = 0
\][/tex]
- Case 2:
[tex]\[
y^{18} - 1 = 0
\][/tex]
Adding 1 to both sides, we get:
[tex]\[
y^{18} = 1
\][/tex]
Taking the 18th root of both sides, we get:
[tex]\[
y = 1
\][/tex]
6. Consider other possible solutions:
[tex]\( y^{18} = 1 \)[/tex] also has other complex solutions because the 18th roots of unity include 18 different complex numbers (including 1). However, for simplicity and focusing on the primary real solutions, we stick with [tex]\( y = 1 \)[/tex].
### Conclusion:
The equation [tex]\( y^{20} = y^2 \)[/tex] is satisfied when [tex]\( y \)[/tex] is either [tex]\( 0 \)[/tex] or [tex]\( 1 \)[/tex].
So, the complete solution for [tex]\( y \)[/tex] is:
[tex]\[
y = 0 \text{ or } y = 1
\][/tex]
