Let's evaluate each statement one by one for their truth value and provide reasons:
### a. [tex]\(2^0 + 3^0 + 4^0 = 3\)[/tex]
True.
Reason: Any non-zero number raised to the power of 0 is 1. Thus, [tex]\(2^0 = 1\)[/tex], [tex]\(3^0 = 1\)[/tex], and [tex]\(4^0 = 1\)[/tex]. Summing these values gives:
[tex]\[ 2^0 + 3^0 + 4^0 = 1 + 1 + 1 = 3 \][/tex]
### b. [tex]\(9^0 - 1^0 = 0\)[/tex]
True.
Reason: Again, any non-zero number raised to the power of 0 is 1. Thus, [tex]\(9^0 = 1\)[/tex] and [tex]\(1^0 = 1\)[/tex]. So,
[tex]\[ 9^0 - 1^0 = 1 - 1 = 0 \][/tex]
### c. [tex]\(2^0 \times 3^0 \times 4^0 = 24\)[/tex]
False.
Reason: Using the rule that any non-zero number raised to the power of 0 is 1, we get [tex]\(2^0 = 1\)[/tex], [tex]\(3^0 = 1\)[/tex], and [tex]\(4^0 = 1\)[/tex]. Multiplying these values results in:
[tex]\[ 2^0 \times 3^0 \times 4^0 = 1 \times 1 \times 1 = 1 \][/tex]
This is not equal to 24.
### d. [tex]\(\left(3^4\right)^2 = 3^8\)[/tex]
True.
Reason: Using the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we have:
[tex]\[ \left(3^4\right)^2 = 3^{4 \cdot 2} = 3^8 \][/tex]
### e. [tex]\(\left(5^2\right)^3 = 100000\)[/tex]
False.
Reason: Again, using the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ \left(5^2\right)^3 = 5^{2 \cdot 3} = 5^6 \][/tex]
Calculating [tex]\(5^6\)[/tex]:
[tex]\[ 5^6 = 15625 \][/tex]
This is not equal to 100000.
### f. [tex]\(3^2 < 2^3\)[/tex]
False.
Reason: Calculating both sides, we get:
[tex]\[ 3^2 = 9 \quad \text{and} \quad 2^3 = 8 \][/tex]
Since [tex]\(9\)[/tex] is not less than [tex]\(8\)[/tex], the statement is false.
### g. [tex]\(\left(16^0 + 12^0\right)\left(16^0 + 12^0\right) > 8^2\)[/tex]
False.
Reason: Using the rule that any non-zero number raised to the power of 0 is 1, we get:
[tex]\[ 16^0 = 1 \quad \text{and} \quad 12^0 = 1 \][/tex]
Thus,
[tex]\[ \left(16^0 + 12^0\right) = 1 + 1 = 2 \][/tex]
Therefore,
[tex]\[ (2) \times (2) = 4 \][/tex]
\- Comparing this with [tex]\(8^2\)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
\- So the expression [tex]\(4\)[/tex] is not greater than [tex]\(64\)[/tex], making the statement false.
In summary, the truth values are:
a. True
b. True
c. False
d. True
e. False
f. False
g. False
