Let's solve each part step-by-step.
### Part (a)
Simplify the expression [tex]\(\frac{2^1 \times 2^5 \times 2^8}{2^2 \times 2^4}\)[/tex]:
1. Combine the powers in the numerator:
[tex]\[
2^1 \times 2^5 \times 2^8 = 2^{1+5+8} = 2^{14}
\][/tex]
2. Combine the powers in the denominator:
[tex]\[
2^2 \times 2^4 = 2^{2+4} = 2^{6}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{2^{14}}{2^{6}} = 2^{14-6} = 2^{8}
\][/tex]
Therefore, the simplified form is:
[tex]\[
2^{8} = 256
\][/tex]
### Part (b)
Simplify the expression [tex]\(\frac{3^6 \times 3^2 \times 3^{-3}}{3^2 \times 3}\)[/tex]:
1. Combine the powers in the numerator:
[tex]\[
3^6 \times 3^2 \times 3^{-3} = 3^{6+2-3} = 3^{5}
\][/tex]
2. Combine the powers in the denominator:
[tex]\[
3^2 \times 3 = 3^2 \times 3^1 = 3^{2+1} = 3^{3}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{3^{5}}{3^{3}} = 3^{5-3} = 3^{2}
\][/tex]
Therefore, the simplified form is:
[tex]\[
3^{2} = 9
\][/tex]
### Final Answer
(a) [tex]\(\frac{2^1 \times 2^5 \times 2^8}{2^2 \times 2^4} = 256\)[/tex]
(b) [tex]\(\frac{3^6 \times 3^2 \times 3^{-3}}{3^2 \times 3} = 9\)[/tex]
