To find the value of [tex]\(4^x \cdot 8^y\)[/tex] given the equation [tex]\(2x + 3y = 11\)[/tex], we will proceed step-by-step:
1. Rewrite the equation in terms of [tex]\(x\)[/tex] or [tex]\(y\)[/tex]:
Since we have one equation with two variables, we need to express one variable in terms of the other. Let's solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[
2x + 3y = 11
\][/tex]
Rearrange to solve for [tex]\(y\)[/tex]:
[tex]\[
3y = 11 - 2x
\][/tex]
[tex]\[
y = \frac{11 - 2x}{3}
\][/tex]
2. Express [tex]\(4^x \cdot 8^y\)[/tex] using powers of 2:
- Note that [tex]\(4 = 2^2\)[/tex] and [tex]\(8 = 2^3\)[/tex].
[tex]\[
4^x = (2^2)^x = 2^{2x}
\][/tex]
[tex]\[
8^y = (2^3)^y = 2^{3y}
\][/tex]
Thus,
[tex]\[
4^x \cdot 8^y = 2^{2x} \cdot 2^{3y}
\][/tex]
3. Combine the exponents using the property of exponents [tex]\((a^m \cdot a^n = a^{m+n})\)[/tex]:
[tex]\[
4^x \cdot 8^y = 2^{2x + 3y}
\][/tex]
4. Substitute [tex]\(2x + 3y\)[/tex] from the given equation:
From the problem, we know:
[tex]\[
2x + 3y = 11
\][/tex]
Therefore:
[tex]\[
4^x \cdot 8^y = 2^{2x + 3y} = 2^{11}
\][/tex]
Thus, the value of [tex]\(4^x \cdot 8^y\)[/tex] is:
[tex]\[
\boxed{2^{11}}
\][/tex]
Therefore, the correct answer is [tex]\(B) 2^{11}\)[/tex].
