Sure, let's go through each statement step-by-step to evaluate their truth value.
1. [tex]\((-3)^4\)[/tex] is less than [tex]\((-4)^3\)[/tex]:
- [tex]\((-3)^4 = 81\)[/tex]
- [tex]\((-4)^3 = -64\)[/tex]
- Since [tex]\(81\)[/tex] is not less than [tex]\(-64\)[/tex], this statement is False.
2. The reciprocal of [tex]\(\left(\frac{3}{4}\right)^6\)[/tex] is [tex]\(\left(\frac{4}{3}\right)^6\)[/tex]:
- [tex]\(\left(\frac{3}{4}\right)^6 = 0.17803125\)[/tex]
- The reciprocal of [tex]\(0.17803125\)[/tex] is approximately [tex]\(5.618655692729766\)[/tex]
- [tex]\(\left(\frac{4}{3}\right)^6 \approx 5.618655692729765\)[/tex]
- Since these are indeed equal, this statement is True.
3. [tex]\(7.8 \times 10^{12}\)[/tex] is greater than [tex]\(9.9 \times 10^8\)[/tex]:
- [tex]\(7.8 \times 10^{12} = 7800000000000\)[/tex]
- [tex]\(9.9 \times 10^8 = 990000000\)[/tex]
- [tex]\(7800000000000\)[/tex] is indeed greater than [tex]\(990000000\)[/tex], so this statement is True.
4. A negative integer raised to an odd positive power is positive:
- For example, [tex]\((-2)^3 = -8\)[/tex]
- A negative integer raised to an odd positive power remains negative.
- Thus, this statement is False.
5. [tex]\((-11)^4\)[/tex] is positive:
- [tex]\((-11)^4 = 14641\)[/tex]
- This is indeed a positive number.
- Therefore, the statement is True.
6. [tex]\(0^{55} + 1^{55} = 0\)[/tex]:
- [tex]\(0^{55} = 0\)[/tex]
- [tex]\(1^{55} = 1\)[/tex]
- [tex]\(0 + 1 = 1\)[/tex], which does not equal [tex]\(0\)[/tex].
- Hence, this statement is False.
7. [tex]\(77^1 \times 0^{77} = 77\)[/tex]:
- [tex]\(77^1 = 77\)[/tex]
- [tex]\(0^{77} = 0\)[/tex]
- [tex]\(77 \times 0 = 0\)[/tex], not [tex]\(77\)[/tex].
- Therefore, this statement is False.
In conclusion:
1. False
2. True
3. True
4. False
5. True
6. False
7. False
