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Let's begin by simplifying the algebraic expressions step-by-step.

### Part a:
\[
\left( \frac{x^4 b^{-2}}{x^{-6} b^2} \right)^{-1}
\]
- Simplify the expression inside the parentheses:
\[
\frac{x^4 b^{-2}}{x^{-6} b^2} = x^{4 - (-6)} b^{-2 - 2} = x^{10} b^{-4}
\]
- Applying the negative exponent:
\[
\left( x^{10} b^{-4} \right)^{-1} = x^{-10} b^{4}
\]

### Part b:
\[
\frac{2^{14} b^{-3} (2w)^3}{2^{18} b^{-5} w}
\]
- Simplify the terms:
\[
(2w)^3 = 2^3 w^3
\]
Now:
\[
\frac{2^{14} b^{-3} \cdot 2^3 w^3}{2^{18} b^{-5} w} = \frac{2^{14 + 3} b^{-3} w^3}{2^{18} b^{-5} w}
\]
Simplifying further:
\[
\frac{2^{17} b^{-3} w^3}{2^{18} b^{-5} w} = 2^{-1} b^{-3 - (-5)} w^{3-1} = \frac{b^2 w^2}{2}
\]

### Part c:
\[
\frac{2^2 x^7 y^{-4}}{(3^7 x^5 y^2)^3}
\]
First, expand the denominator:
\[
(3^7 x^5 y^2)^3 = 3^{7 \cdot 3} x^{5 \cdot 3} y^{2 \cdot 3} = 3^{21} x^{15} y^6
\]
Now simplify the fraction:
\[
\frac{2^2 x^7 y^{-4}}{3^{21} x^{15} y^6} = \frac{2^2}{3^{21}} x^{7-15} y^{-4-6} = \frac{4}{3^{21}} x^{-8} y^{-10}
\]

### Part d:
\[
\frac{(2 \cdot 3)^4 c^{-5} d^{-9}}{2^7 \cdot 3^{-2} (5c)^{-3}}
\]
First simplify the numerator and denominator:
\[
(2 \cdot 3)^4 = 2^4 \cdot 3^4
\]
Now:
\[
\frac{2^4 \cdot 3^4 c^{-5} d^{-9}}{2^7 \cdot 3^{-2} \cdot 5^{-3} c^{-3}}
\]
Simplify further:
\[
= \frac{2^4 \cdot 3^4 c^{-5} d^{-9}}{2^7 \cdot 3^{-2} \cdot 5^{-3} c^{-3}} = \frac{2^{4-7} \cdot 3^{4-(-2)} c^{-5 - (-3)} d^{-9}}{5^{-3}}
\]
Simplifying exponents:
\[
= \frac{2^{-3} \cdot 3^6 c^{-2} d^{-9}}{5^{-3}} = \frac{3^6 c^{-2} d^{-9} \cdot 5^3}{2^3} = \frac{729 \cdot 125 c^{-2} d^{-9}}{8} = \frac{91125 c^{-2} d^{-9}}{8}
\]

### Exercise 26:
The expression is:
\[
\left( \frac{3a^2b - \frac{ab}{2}}{2} - \frac{-4a^2b + \frac{2ab}{3}}{3} \right) + (a^2b + ab)
\]
I will now simplify this expression to find the coefficient of \(ab\).

Let me simplify this next.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can still guide you through the solution manually if you'd like. Would you like me to do that, or do you have any other questions?

Simplify the following expressions and find the coefficient of 'ab' in the given algebraic expression.

Learn how to simplify complex algebraic expressions with exponents and polynomials. This example covers exponent rules, fraction simplification, and polynomial expansion, and guides you through finding the coefficient of 'ab'. Suitable for students in Grades 10-12.

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