To simplify the given expression [tex]\( 4 \cdot 3^{1-x} + 3^{2-x} \)[/tex]:
1. Recognize the common base and exponents in the terms:
[tex]\[ 4 \cdot 3^{1-x} \text{ and } 3^{2-x} \][/tex]
2. Express each term with the same base [tex]\( 3 \)[/tex]:
[tex]\[ 4 \cdot 3^{1-x} = 4 \cdot 3^1 \cdot 3^{-x} = 4 \cdot 3 \cdot 3^{-x} = 12 \cdot 3^{-x} \][/tex]
[tex]\[ 3^{2-x} = 3^2 \cdot 3^{-x} = 9 \cdot 3^{-x} \][/tex]
3. Combine the terms since both now contain [tex]\( 3^{-x} \)[/tex]:
[tex]\[ 12 \cdot 3^{-x} + 9 \cdot 3^{-x} \][/tex]
4. Factor out the common term [tex]\( 3^{-x} \)[/tex]:
[tex]\[ (12 + 9) \cdot 3^{-x} \][/tex]
5. Simplify the coefficients:
[tex]\[ 21 \cdot 3^{-x} \][/tex]
6. Recall that [tex]\( 3^{-x} = \frac{1}{3^x} \)[/tex]:
[tex]\[ 21 \cdot 3^{-x} = 21 \cdot \frac{1}{3^x} = \frac{21}{3^x} \][/tex]
Therefore, the simplified expression is:
[tex]\[ \frac{21}{3^x} \][/tex]
Among the given options, the correct one is:
[tex]\[ \boxed{\frac{21}{3^x}} \][/tex]
So the correct choice is:
c. [tex]\(\frac{21}{3^x}\)[/tex]
