The problem appears to consist of evaluating mathematical expressions involving powers of decimals. I will solve each part one by one:

Problem Breakdown

1. $a) \ 0.04^{2/3}$
2. $b) \ 0.16^{-1/2}$
3. $c) \ 0.027^{3/2}$
4. $d) \ 0.0016^{-4/3}$
5. $e) \ 0.0625^{-4/5}$
6. $f) \ 0.0081^{-1.25}$
7. $g) \ 0.000032^{0.375}$

I'll compute these now.import sympy as sp

Define each expression to evaluate

expressions = { "a": (0.04)(2/3), "b": (0.16)(-1/2), "c": (0.027)(3/2), "d": (0.0016)(-4/3), "e": (0.0625)(-4/5), "f": (0.0081)(-1.25), "g": (0.000032)**(0.375), }

Evaluate each expression

results = {key: sp.N(value) for key, value in expressions.items()} results{'a': 0.116960709528515, 'b': 2.50000000000000, 'c': 0.00443655271579185, 'd': 5343.67483336468, 'e': 9.18958683997628, 'f': 411.522633744856, 'g': 0.0206267707544249}{'a': 0.116960709528515, 'b': 2.50000000000000, 'c': 0.00443655271579185, 'd': 5343.67483336468, 'e': 9.18958683997628, 'f': 411.522633744856, 'g': 0.0206267707544249}successHere are the computed results for each part:

1. $a) \ 0.04^{2/3} = 0.11696$
2. $b) \ 0.16^{-1/2} = 2.5$
3. $c) \ 0.027^{3/2} = 0.00444$
4. $d) \ 0.0016^{-4/3} = 5343.67$
5. $e) \ 0.0625^{-4/5} = 9.19$
6. $f) \ 0.0081^{-1.25} = 411.52$
7. $g) \ 0.000032^{0.375} = 0.02063$

If you would like a step-by-step explanation for any of these calculations, let me know!

Related Questions:

1. How are fractional exponents evaluated step-by-step?
2. What does it mean when an exponent is negative?
3. Why are some results very large or very small?
4. How can these calculations be visualized with graphs?
5. How do these operations relate to roots (e.g., square or cube roots)?

Tip:

When working with fractional exponents, remember that the numerator indicates the power, and the denominator represents the root. For example, $x^{2/3} = (x^2)^{1/3}$.