To determine which choice is equivalent to the expression \(2^{6.38}\), we need to break down the original expression and see how it can be rewritten using the properties of exponents.
The expression \(2^{6.38}\) is given in an exponential form. Let's analyze each given choice:
Choice A:
[tex]\[
2^6 \cdot 2^{3 / 10} \cdot 2^{8 / 100}
\][/tex]
Using the properties of exponents, specifically \(a^m \cdot a^n = a^{m+n}\), we can combine these terms:
[tex]\[
2^6 \cdot 2^{0.3} \cdot 2^{0.08} = 2^{6 + 0.3 + 0.08} = 2^{6.38}
\][/tex]
So, this choice is a possible match.
Choice B:
[tex]\[
2^{6 + 3 / 10 + 8 / 10}
\][/tex]
Here, we need to simplify the exponents inside the base 2:
[tex]\[
2^{6 + 0.3 + 0.8} = 2^{6 + 1.1} = 2^{7.1}
\][/tex]
This is not equivalent to \(2^{6.38}\).
Choice C:
[tex]\[
2^6 + 2^{3 / 10} + 2^{8 / 100}
\][/tex]
This expression is a sum of three different powers of 2, not a single exponential term. Therefore, it cannot be equivalent to \(2^{6.38}\).
Choice D:
[tex]\[
2^6 \cdot 2^{38 / 10}
\][/tex]
Simplifying the exponent inside the base 2:
[tex]\[
2^6 \cdot 2^{3.8} = 2^{6 + 3.8} = 2^{9.8}
\][/tex]
This is also not equivalent to \(2^{6.38}\).
After evaluating each option, we see that Choice A:
[tex]\[
2^6 \cdot 2^{3 / 10} \cdot 2^{8 / 100}
\][/tex]
is indeed equivalent to the expression \(2^{6.38}\). Therefore, the correct choice is:
A. [tex]\(2^6 \cdot 2^{3 / 10} \cdot 2^{8 / 100}\)[/tex].
