To determine which choice is equivalent to the expression [tex]\(2^{7.19}\)[/tex], we need to simplify [tex]\(2^{7.19}\)[/tex] and compare it with each option.
First, let’s rewrite [tex]\(2^{7.19}\)[/tex]:
[tex]\[ 2^{7.19} = 2^{7 + 0.19} \][/tex]
The exponent [tex]\(7.19\)[/tex] can be broken down into [tex]\(7\)[/tex] and [tex]\(0.19\)[/tex]. Then we can use the property of exponents that [tex]\(a^{b+c} = a^b \cdot a^c\)[/tex]:
[tex]\[ 2^{7.19} = 2^7 \cdot 2^{0.19} \][/tex]
Next, we need to analyze the provided choices:
Choice A:
[tex]\[
2^7 \cdot 2^{1/10} \cdot 2^{9/100}
\][/tex]
We can combine the exponents using the product of powers property:
[tex]\[
2^7 \cdot 2^{1/10} \cdot 2^{9/100} = 2^7 \cdot 2^{10/100} \cdot 2^{9/100} = 2^7 \cdot 2^{(10+9)/100} = 2^7 \cdot 2^{19/100} = 2^{7 + 0.19}
\][/tex]
This matches our expression for [tex]\(2^{7.19}\)[/tex], indicating that choice A is an equivalent expression.
Choice B:
[tex]\[
2^7 \cdot 2^{19/10}
\][/tex]
Combine the exponents:
[tex]\[
2^7 \cdot 2^{19/10} = 2^{7 + 19/10} = 2^{7 + 1.9}
\][/tex]
This simplifies to [tex]\(2^{8.9}\)[/tex], which is not the same as [tex]\(2^{7.19}\)[/tex].
Choice C:
[tex]\[
2^{7 + 1/10 + 9/10}
\][/tex]
Simplify the combined exponent:
[tex]\[
2^{7 + 1/10 + 9/10} = 2^{7 + 1} = 2^8
\][/tex]
This simplifies to [tex]\(2^8\)[/tex], which is not the same as [tex]\(2^{7.19}\)[/tex].
Choice D:
[tex]\[
2^7 + 2^{1/10} + 2^{9/100}
\][/tex]
This is a sum of three terms, not a product. The expression [tex]\(2^{7.19}\)[/tex] cannot be represented as the sum of exponential terms.
Based on the simplification and comparison, the equivalent expression to [tex]\(2^{7.19}\)[/tex] is:
[tex]\[
\boxed{A}
\][/tex]
