To solve for [tex]\( 4^x \cdot 8^y \)[/tex] given the equation [tex]\( 2x + 3y = 11 \)[/tex], we can break down the problem step-by-step.
1. Rewrite the Expression in Terms of Base 2:
- Note that [tex]\( 4 \)[/tex] can be expressed as [tex]\( 2^2 \)[/tex].
- Similarly, [tex]\( 8 \)[/tex] can be expressed as [tex]\( 2^3 \)[/tex].
2. Express [tex]\( 4^x \)[/tex] and [tex]\( 8^y \)[/tex] Using Base 2:
- [tex]\( 4^x = (2^2)^x = 2^{2x} \)[/tex]
- [tex]\( 8^y = (2^3)^y = 2^{3y} \)[/tex]
3. Combine the Expressions:
- The product [tex]\( 4^x \cdot 8^y = 2^{2x} \cdot 2^{3y} \)[/tex].
4. Use the Property of Exponents:
- When multiplying with the same base, you add the exponents: [tex]\( 2^{2x} \cdot 2^{3y} = 2^{2x + 3y} \)[/tex].
5. Substitute From the Given Equation:
- We know that [tex]\( 2x + 3y = 11 \)[/tex]. Therefore, [tex]\( 2x + 3y \)[/tex] can be replaced with 11.
6. Final Expression:
- Substituting in the exponents, we get [tex]\( 2^{2x + 3y} = 2^{11} \)[/tex].
Thus, the value of [tex]\( 4^x \cdot 8^y \)[/tex] is [tex]\( 2^{11} \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{2^{11}} \][/tex]
