Let's analyze each expression step-by-step to determine which of them are equal to 1.
1. Expression: [tex]\(0^1\)[/tex]
For any base [tex]\(a \)[/tex] raised to the power of 1 ([tex]\( a^1 \)[/tex]), the result is [tex]\(a\)[/tex].
Therefore, [tex]\(0^1 = 0\)[/tex].
This expression is not equal to 1.
2. Expression: [tex]\(1291^0\)[/tex]
Any non-zero number raised to the power of 0 is equal to 1.
Therefore, [tex]\(1291^0 = 1\)[/tex].
This expression is equal to 1.
3. Expression: [tex]\(-\left(\frac{5}{8}\right)^0\)[/tex]
Any non-zero number raised to the power of 0 is equal to 1.
Therefore, [tex]\(\left(\frac{5}{8}\right)^0 = 1\)[/tex].
Taking the negative sign into account, we have [tex]\(-1\)[/tex].
This expression is not equal to 1 because it equals -1.
4. Expression: [tex]\(-(-x)^0\)[/tex]
For any number [tex]\(x \)[/tex], [tex]\((-x)^0\)[/tex] is equal to 1.
Taking the negative sign into account, we have [tex]\(-1\)[/tex].
This expression is not equal to 1 because it equals -1.
5. Expression: [tex]\(\left(-\frac{d}{t}\right)^0\)[/tex]
Any non-zero number raised to the power of 0 is equal to 1.
Therefore, [tex]\(\left(-\frac{d}{t}\right)^0 = 1\)[/tex].
This expression is equal to 1.
6. Expression: [tex]\(\left(\frac{1}{3}\right)^0\)[/tex]
Any non-zero number raised to the power of 0 is equal to 1.
* Therefore, [tex]\(\left(\frac{1}{3}\right)^0 = 1\)[/tex].
This expression is equal to 1.
To summarize:
- The expressions that are equal to 1 are: [tex]\(1291^0\)[/tex], [tex]\(\left(-\frac{d}{t}\right)^0\)[/tex], and [tex]\(\left(\frac{1}{3}\right)^0\)[/tex].
So, the expressions equal to 1 are:
[tex]\[ \boxed{2, 3, 5, 6} \][/tex]
