Let's solve the given expression step by step:
The expression is:
[tex]\[ J = \left[ \left(2^4\right)^2 \right]^{-1} \cdot 2^{3^2} + 3^{4^0} \][/tex]
Step 1: Calculate [tex]\(\left(2^4\right)^2\)[/tex]
First, calculate [tex]\(2^4\)[/tex]:
[tex]\[ 2^4 = 16 \][/tex]
Now, raise this result to the power of 2:
[tex]\[ 16^2 = 256 \][/tex]
So, [tex]\(\left(2^4\right)^2 = 256\)[/tex].
Step 2: Find the inverse of 256
The inverse of a number [tex]\(x\)[/tex] is [tex]\(x^{-1}\)[/tex]:
[tex]\[ \left(256\right)^{-1} = \frac{1}{256} = 0.00390625 \][/tex]
Step 3: Calculate [tex]\(2^{3^2}\)[/tex]
First, calculate [tex]\(3^2\)[/tex]:
[tex]\[ 3^2 = 9 \][/tex]
Now, raise 2 to the power of 9:
[tex]\[ 2^9 = 512 \][/tex]
So, [tex]\(2^{3^2} = 512\)[/tex].
Step 4: Multiply the inverse of 256 by [tex]\(2^{3^2}\)[/tex]
[tex]\[ \left(256^{-1}\right) \cdot 2^{9} = 0.00390625 \cdot 512 = 2 \][/tex]
Step 5: Calculate [tex]\(3^{4^0}\)[/tex]
First, calculate [tex]\(4^0\)[/tex]:
[tex]\[ 4^0 = 1 \][/tex]
Now, raise 3 to the power of 1:
[tex]\[ 3^1 = 3 \][/tex]
So, [tex]\(3^{4^0} = 3\)[/tex].
Step 6: Add the results from steps 4 and 5
[tex]\[ 2 + 3 = 5 \][/tex]
So the value of [tex]\(J\)[/tex] is:
[tex]\[ J = 5 \][/tex]
Therefore, the correct answer is:
d) 5
