Which of the following is an equivalent expression to [tex]8 \cdot(1+0.08)^{2x}[/tex]?

A. [tex]64 \cdot(1.08)^x[/tex]
B. [tex]16 \cdot(1.08)^x[/tex]
C. [tex]8 \cdot(1.1664)^x[/tex]
D. [tex]8 \cdot(1.0064)^x[/tex]



Answer :

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]