Let's solve the given expression step-by-step.
The expression given is:
[tex]\[ 8 \cdot (1 + 0.08)^{2x} \][/tex]
First, simplify the base inside the parenthesis:
[tex]\[ 1 + 0.08 = 1.08 \][/tex]
Now the expression becomes:
[tex]\[ 8 \cdot (1.08)^{2x} \][/tex]
To make a comparison with the given options, let's consider rewriting the base in different forms to match an equivalent expression.
Let's test the exponent manipulation. We know from exponent rules that:
[tex]\[ (a^b)^c = a^{bc} \][/tex]
If we set:
[tex]\[ 1.08^{2x} \][/tex]
we can raise 1.08 to the power of 2:
[tex]\[ 1.08^2 \approx 1.1664 \][/tex]
Therefore:
[tex]\[ (1.08^2)^x = 1.1664^x \][/tex]
Thus, multiplying this by 8, we get:
[tex]\[ 8 \cdot (1.1664)^x \][/tex]
Hence, the equivalent expression to [tex]\( 8 \cdot (1 + 0.08)^{2x} \)[/tex] is:
[tex]\[ 8 \cdot (1.1664)^x \][/tex]
So the correct answer is:
[tex]\[ 8 \cdot (1.1664)^x \][/tex]
Therefore, the option that matches is:
[tex]\[ \boxed{8 \cdot (1.1664)^x} \][/tex]
