To solve the equation [tex]\( t^3 = (t^2) \cdot t = -1 \)[/tex], we need to find a value of [tex]\( t \)[/tex] that satisfies this condition. Let's proceed with a step-by-step reasoning:
1. Starting with the equation:
[tex]\[
t^3 = (t^2) \cdot t = -1
\][/tex]
2. Understanding the equation:
- [tex]\( (t^2) \cdot t \)[/tex] means the square of [tex]\( t \)[/tex] multiplied by [tex]\( t \)[/tex] itself, which simplifies to [tex]\( t^3 \)[/tex].
- We need [tex]\( t^3 \)[/tex] to equal [tex]\(-1\)[/tex].
3. Exploring possible values for [tex]\( t \)[/tex]:
We seek a value of [tex]\( t \)[/tex] such that:
[tex]\[
t^3 = -1
\][/tex]
4. Trial and specific value: [tex]\( t = -1 \)[/tex]:
Let's check [tex]\( t = -1 \)[/tex]:
[tex]\[
t = -1
\][/tex]
5. Compute [tex]\( t^2 \)[/tex] when [tex]\( t = -1 \)[/tex]:
[tex]\[
t^2 = (-1)^2 = 1
\][/tex]
6. Compute [tex]\( t^3 \)[/tex] when [tex]\( t = -1 \)[/tex]:
[tex]\[
t^3 = (-1)^3 = -1
\][/tex]
7. Verification:
- Substituting [tex]\( t = -1 \)[/tex] back into the original equation:
[tex]\[
t^3 = (t^2) \cdot t
\][/tex]
- Using the values we computed:
[tex]\[
-1 = 1 \cdot (-1) = -1
\][/tex]
- The values are consistent with the original condition.
Therefore, the value of [tex]\( t \)[/tex] that satisfies the equation [tex]\( t^3 = (t^2) \cdot t = -1 \)[/tex] is [tex]\( t = -1 \)[/tex].
Summary:
- [tex]\( t = -1 \)[/tex]
- [tex]\( t^2 = 1 \)[/tex]
- [tex]\( t^3 = -1 \)[/tex]
Thus, the solution to the problem is:
[tex]\[
(t, t^2, t^3) = (-1, 1, -1)
\][/tex]
