Certainly! Let's go through the simplification step-by-step.
Given the expression:
[tex]\[ \frac{4^x \cdot 2^{x+1}}{2^{2x+1} \cdot 2^{x+2}} \][/tex]
1. Rewrite [tex]\(4^x\)[/tex] as an expression with base 2:
[tex]\[ 4^x = (2^2)^x = 2^{2x} \][/tex]
Substituting [tex]\(4^x\)[/tex] with [tex]\(2^{2x}\)[/tex], the expression becomes:
[tex]\[ \frac{2^{2x} \cdot 2^{x+1}}{2^{2x+1} \cdot 2^{x+2}} \][/tex]
2. Combine the exponents in both the numerator and the denominator using the property [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
- For the numerator:
[tex]\[ 2^{2x} \cdot 2^{x+1} = 2^{2x + (x+1)} = 2^{3x+1} \][/tex]
- For the denominator:
[tex]\[ 2^{2x+1} \cdot 2^{x+2} = 2^{(2x+1) + (x+2)} = 2^{3x+3} \][/tex]
So now the expression is:
[tex]\[ \frac{2^{3x+1}}{2^{3x+3}} \][/tex]
3. Simplify the expression by subtracting the exponents in the denominator from the exponents in the numerator (using the property [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]):
[tex]\[ 2^{(3x+1) - (3x+3)} = 2^{3x+1 - 3x-3} = 2^{-2} \][/tex]
4. Simplify the exponent [tex]\(2^{-2}\)[/tex]:
[tex]\[ 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \][/tex]
Therefore, the fully simplified expression is:
[tex]\[ \frac{1}{4} = 0.25 \][/tex]
So, the result of the given expression is:
[tex]\[ \boxed{0.25} \][/tex]
