Let's work through some of the problems from the image you uploaded. I'll solve a few of them step by step:

Problem 23:

$\frac{-4w^{12}}{12w^3}$

Solution:

1. Simplify the coefficients: $\frac{-4}{12} = -\frac{1}{3}$
2. Apply the quotient rule for exponents: $\frac{w^{12}}{w^3} = w^{12-3} = w^9$
3. Final expression: $\frac{-4w^{12}}{12w^3} = -\frac{1}{3}w^9$

Problem 24:

$\frac{13r^7}{39r^4}$

Solution:

1. Simplify the coefficients: $\frac{13}{39} = \frac{1}{3}$
2. Apply the quotient rule for exponents: $\frac{r^7}{r^4} = r^{7-4} = r^3$
3. Final expression: $\frac{13r^7}{39r^4} = \frac{1}{3}r^3$

Problem 25:

$\left(\frac{2a^4c^3}{5b^2d^2}\right)^2$

Solution:

1. Square the numerator and denominator: $\frac{(2a^4c^3)^2}{(5b^2d^2)^2}$
2. Expand the powers: $= \frac{2^2 a^{4\cdot 2} c^{3\cdot 2}}{5^2 b^{2\cdot 2} d^{2\cdot 2}}$ $= \frac{4a^8c^6}{25b^4d^4}$
3. Final expression: $\frac{4a^8c^6}{25b^4d^4}$

Problem 26:

$\frac{m^6}{m^2n^3}$

Solution:

1. Apply the quotient rule for exponents to the $m$ terms: $\frac{m^6}{m^2} = m^{6-2} = m^4$
2. The $n^3$ remains in the denominator: $= \frac{m^4}{n^3}$
3. Final expression: $\frac{m^4}{n^3}$

Problem 27:

$\left(\frac{24a^{11}b^{16}c^6}{18a^6b^6c^6}\right)^3$

Solution:

1. Simplify the coefficients: $\frac{24}{18} = \frac{4}{3}$
2. Simplify the exponents using the quotient rule for each variable:
   • $a^{11}/a^6 = a^{11-6} = a^5$
   • $b^{16}/b^6 = b^{16-6} = b^{10}$
   • $c^6/c^6 = 1$
3. Simplified fraction: $\frac{4a^5b^{10}}{3}$
4. Apply the cube (raise everything to the power of 3): $\left(\frac{4a^5b^{10}}{3}\right)^3 = \frac{4^3a^{5\cdot 3}b^{10\cdot 3}}{3^3} = \frac{64a^{15}b^{30}}{27}$
5. Final expression: $\frac{64a^{15}b^{30}}{27}$

Let me know if you'd like more details or if you'd like to continue with other problems.

Related questions:

1. How do you apply the quotient rule when simplifying fractions with exponents?
2. What is the general approach to simplifying expressions involving powers of fractions?
3. How do you handle coefficients when raising a fraction to a power?
4. How can we simplify complex rational expressions with multiple variables?
5. How do you manage exponents when variables in the numerator and denominator have the same base?

Tip: Always remember to simplify coefficients separately from the variables when simplifying rational expressions.