Let's analyze each statement in detail:
1. [tex]\(\left(5^2\right)^{10}\)[/tex] equals [tex]\(5^2 \times 10\)[/tex]:
- According to the rules of exponents, [tex]\(\left(a^m\right)^n = a^{mn}\)[/tex]. So, [tex]\(\left(5^2\right)^{10} = 5^{2 \times 10} = 5^{20}\)[/tex].
- On the other hand, [tex]\(5^2 \times 10 = 25 \times 10 = 250\)[/tex].
- Clearly, [tex]\(5^{20}\)[/tex] is not equal to 250.
- Therefore, this statement is False.
2. The values of [tex]\(100^1\)[/tex] and [tex]\(1^{100}\)[/tex] are equal:
- Calculate [tex]\(100^1\)[/tex]: which equals 100.
- Calculate [tex]\(1^{100}\)[/tex]: Since any number to the power of 0 is 1, [tex]\(1^{100} = 1\)[/tex].
- 100 is not equal to 1.
- Therefore, this statement is False.
3. [tex]\(\left(\frac{11^5}{11^4}\right)^2\)[/tex] is 121:
- Simplify [tex]\(\frac{11^5}{11^4}\)[/tex]: According to the rules of exponents, [tex]\(a^m / a^n = a^{m - n}\)[/tex]. Thus, [tex]\(\frac{11^5}{11^4} = 11^{5-4} = 11\)[/tex].
- Now calculate [tex]\(\left(11\right)^2\)[/tex]: [tex]\(11^2 = 121\)[/tex].
- Therefore, this statement is True.
4. The value of [tex]\(6^{-2}\)[/tex] is -36:
- According to the rules of exponents, [tex]\(a^{-m} = \frac{1}{a^m}\)[/tex]. Thus, [tex]\(6^{-2} = \frac{1}{6^2} = \frac{1}{36}\)[/tex].
- [tex]\(\frac{1}{36}\)[/tex] is not equal to -36.
- Therefore, this statement is False.
5. [tex]\(\left(\frac{1}{4}\right)^{-8} \div \left(\frac{1}{4}\right)^{-8}\)[/tex] is less than 1:
- According to the rules of exponents, any non-zero number divided by itself equals 1. Therefore, [tex]\(\frac{\left(\frac{1}{4}\right)^{-8}}{\left(\frac{1}{4}\right)^{-8}} = 1\)[/tex].
- 1 is not less than 1.
- Therefore, this statement is False.
