Let's go through each part of the question step by step. The instructions ask us to solve the expressions without using tools and then verify the result using CAS (Computer Algebra System).

Problem 1.59

a) $\frac{(2x^{-5})^{-3}}{8^{-1}}$

We'll simplify this by applying the properties of exponents:

1. Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$. $(2x^{-5})^{-3} = 2^{-3} \cdot x^{15} = \frac{1}{8} \cdot x^{15}$
2. Now, simplify the denominator $8^{-1}$: $8^{-1} = \frac{1}{8}$
3. The expression now becomes: $\frac{\frac{1}{8}x^{15}}{\frac{1}{8}} = x^{15}$

So, the answer for part (a) is $x^{15}$.

b) $\frac{16x^2 \cdot (6x)^{-2}}{(3x^2)^{-3}}$

1. Simplify the terms separately: $16x^2 \cdot (6x)^{-2} = 16x^2 \cdot \frac{1}{36x^2} = \frac{16x^2}{36x^2} = \frac{16}{36} = \frac{4}{9}$
2. Simplify the denominator: $(3x^2)^{-3} = \frac{1}{(3x^2)^3} = \frac{1}{27x^6}$
3. The expression becomes: $\frac{\frac{4}{9}}{\frac{1}{27x^6}} = \frac{4}{9} \cdot \frac{27x^6}{1} = \frac{4 \cdot 27x^6}{9} = 12x^6$

So, the answer for part (b) is $12x^6$.

c) $\left( \frac{x^{-2}}{1000} \right)^{-1} \cdot \frac{(2x^3)^{-1}}{20x \cdot (5x^{-1})^2}$

1. Simplify the first part: $\left( \frac{x^{-2}}{1000} \right)^{-1} = \frac{1000}{x^{-2}} = 1000x^2$
2. Simplify the second part: $\frac{(2x^3)^{-1}}{20x \cdot (5x^{-1})^2}$ First, simplify $(2x^3)^{-1}$: $(2x^3)^{-1} = \frac{1}{2x^3}$ Then, simplify $(5x^{-1})^2$: $(5x^{-1})^2 = 25x^{-2}$ Now the expression becomes: $\frac{\frac{1}{2x^3}}{20x \cdot 25x^{-2}} = \frac{\frac{1}{2x^3}}{500x^{-1}} = \frac{1}{2x^3} \cdot \frac{x}{500} = \frac{x}{1000x^3} = \frac{1}{1000x^2}$
3. Multiply both parts together: $1000x^2 \cdot \frac{1}{1000x^2} = 1$

So, the answer for part (c) is $1$.

Would you like more details on any step, or do you have any specific questions?

Here are 5 related questions to deepen your understanding:

1. What is the general rule for raising a power to another power?
2. How do negative exponents affect the base in expressions like $a^{-n}$?
3. How can the property of exponents simplify multiplication of terms with the same base?
4. Why does dividing by a fraction involve multiplying by its reciprocal?
5. How do fractional exponents relate to roots in expressions?

Tip: Whenever dealing with exponents and powers, it helps to first apply the rules of exponents systematically before simplifying the terms.