To determine whether [tex]\((-1)^6\)[/tex] is equivalent to [tex]\(-1 \times -1 \times -1 \times -1 \times -1 \times -1\)[/tex], we need to carefully evaluate the expression step-by-step.
First, let's interpret [tex]\((-1)^6\)[/tex]:
[tex]\[
(-1)^6 \text{ means that } -1 \text{ is multiplied by itself 6 times.}
\][/tex]
Now, let's explicitly write out the multiplication:
[tex]\[
(-1) \times (-1) \times (-1) \times (-1) \times (-1) \times (-1)
\][/tex]
Next, we can group the multiplications in pairs for easier computation:
[tex]\[
((-1) \times (-1)) \times ((-1) \times (-1)) \times ((-1) \times (-1))
\][/tex]
We know that multiplying two negative numbers results in a positive number:
[tex]\[
(-1) \times (-1) = 1
\][/tex]
So, substituting back we get:
[tex]\[
1 \times 1 \times 1 = 1
\][/tex]
Therefore,
[tex]\[
(-1) \times (-1) \times (-1) \times (-1) \times (-1) \times (-1) = 1
\][/tex]
Now we compare this result to [tex]\((-1)^6\)[/tex]:
[tex]\((-1)^6\)[/tex] also simplifies to:
[tex]\[
(-1)^6 = 1
\][/tex]
So, yes, [tex]\((-1)^6 = 1\)[/tex] is equivalent to [tex]\(-1 \times -1 \times -1 \times -1 \times -1 \times -1\)[/tex].
Thus, the statement is \textbf{True}.
