Let's simplify the given mathematical expressions step-by-step.
### Part (i)
Given expression:
[tex]\[
\frac{5^{-2} \times 3^{-3} \times(125)^{\frac{2}{3}}}{\frac{-2}{-1}}
\][/tex]
First, simplify the denominator:
[tex]\[
\frac{-2}{-1} = \frac{2}{1} = 2
\][/tex]
Now, combining the numerator:
[tex]\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\][/tex]
[tex]\[
3^{-3} = \frac{1}{3^3} = \frac{1}{27}
\][/tex]
[tex]\[
125^{\frac{2}{3}} = (5^3)^{\frac{2}{3}} = 5^2 = 25
\][/tex]
Thus, the numerator becomes:
[tex]\[
\frac{1}{25} \times \frac{1}{27} \times 25 = \frac{25}{25 \times 27} = \frac{1}{27}
\][/tex]
Now, the full expression is:
[tex]\[
\frac{\frac{1}{27}}{2} = \frac{1}{27} \times \frac{1}{2} = \frac{1}{54}
\][/tex]
Next, evaluate the second part involving subtraction:
[tex]\[
(27)^3 \times (32)^2
\][/tex]
Calculating individual terms:
[tex]\[
27^3 = 19683
\][/tex]
[tex]\[
32^2 = 1024
\][/tex]
Thus, multiplying these:
[tex]\[
19683 \times 1024 = 20155392
\][/tex]
Now the first part simplifies to:
[tex]\[
\frac{1/54}{20155392} = \frac{1}{1098391488}
\][/tex]
### Part (ii)
Given expression:
[tex]\[
64^{\frac{-1}{6}} \times 216^{\frac{-1}{3}} \times (81)^{\frac{1}{4}}
\][/tex]
Simplifying each term:
[tex]\[
64^{\frac{-1}{6}} = \left(2^6\right)^{\frac{-1}{6}} = 2^{-1} = \frac{1}{2}
\][/tex]
[tex]\[
216^{\frac{-1}{3}} = \left(6^3\right)^{\frac{-1}{3}} = 6^{-1} = \frac{1}{6}
\][/tex]
[tex]\[
81^{\frac{1}{4}} = \left(3^4\right)^{\frac{1}{4}} = 3
\][/tex]
Combining these terms:
[tex]\[
\frac{1}{2} \times \frac{1}{6} \times 3 = \frac{3}{12} = 0.25
\][/tex]
### Part (iii)
[tex]\[
\frac{(512)^{-\frac{1}{2}} \times (16)^{\frac{1}{4}} \times (9)^{-\frac{1}{2}}}{(512)^2 \times (16)^4 \times (9)^2}
\][/tex]
Simplifying the numerator:
[tex]\[
(512)^{-\frac{1}{2}} = \left(8^3\right)^{-\frac{1}{2}} \approx 0.03125
\][/tex]
[tex]\[
(16)^{\frac{1}{4}} = 2
\][/tex]
[tex]\[
(9)^{-\frac{1}{2}} = \frac{1}{3}
\][/tex]
Combining these:
[tex]\[
0.03125 \times 2 \times \frac{1}{3} \approx 0.02084
\][/tex]
Simplifying the denominator:
[tex]\[
512^2 = 262144
\][/tex]
[tex]\[
16^4 = 2^{16} = 65536
\][/tex]
[tex]\[
9^2 = 81
\][/tex]
Combining these:
[tex]\[
262144 \times 65536 \times 81 \approx 1391569403904
\][/tex]
Finally, combining the result:
[tex]\[
\frac{0.02084}{1391569403904} \approx 1.497 \times 10^{-11}
\][/tex]
Collating the results of each part:
[tex]\[
(expr1, expr2, result\_part1) = (\frac{1}{27}, 20155392, 9.187873159955666 \times 10^{-10})
\][/tex]
[tex]\[
(expr3, expr4, expr5, result\_part2) = (0.25, 0.025, 1391569403904, 6.097634333173326 \times 10^{-12})
\][/tex]
