Simplify: 1. 8x^2 / 12x^4; 2. 24x^2y / 32xy^3; 3. (x + 1)^3 / (x + 1)^5; 4. (3x - 2)^6 / (3x - 2)^4; 5. (2n - 3) / (3 - 2n); 6. 15y^2(y + 3) / 20y^3(y + 3); 7. 15y(y + 1) / 45y^2(y... Simplify: 1. 8x^2 / 12x^4; 2. 24x^2y / 32xy^3; 3. (x + 1)^3 / (x + 1)^5; 4. (3x - 2)^6 / (3x - 2)^4; 5. (2n - 3) / (3 - 2n); 6. 15y^2(y + 3) / 20y^3(y + 3); 7. 15y(y + 1) / 45y^2(y + 1); 8. 24x^3(4 - x) / 18x(4 - x); 9. 6x^3(2 - x) / 12x(2 - x); 10. 12x^3(5 - 2x) / 18x(2 - 5); 11. a^2 + 6a / ac + 6c; 12. x^2 + 4x / 3x + 12; 13. 6 - 8x / 4x^2 - 3x; 14. x^2 + 2x - 15 / x^2 - 10x + 21; 15. x^2 - 2x - 3 / x^2 - 5x - 6 Read more

Steps to Solve

1. Simplifying Expression 1: Start with the expression $$ \frac{8x^2}{12x^4} $$ Factor out the greatest common divisor (GCD), which is 4: $$ \frac{8x^2 \div 4}{12x^4 \div 4} = \frac{2x^2}{3x^4} $$ Now, simplify the powers of $x$: $$ \frac{2}{3} \cdot \frac{x^2}{x^4} = \frac{2}{3} \cdot \frac{1}{x^{4-2}} = \frac{2}{3x^2} $$

2. Simplifying Expression 2: Start with $$ \frac{24x^2y}{32xy^3} $$ The GCD is 8: $$ \frac{24x^2y \div 8}{32xy^3 \div 8} = \frac{3x^2y}{4xy^3} $$ Next, simplify: $$ \frac{3}{4} \cdot \frac{x^2}{x} \cdot \frac{y}{y^3} = \frac{3}{4} \cdot x^{2-1} \cdot y^{1-3} = \frac{3x}{4y^2} $$

3. Simplifying Expression 3: Start with $$ \frac{(x+1)^3}{(x+1)^5} $$ Simplifying the powers: $$ (x+1)^{3-5} = (x+1)^{-2} = \frac{1}{(x+1)^2} $$

4. Simplifying Expression 4: Start with $$ \frac{(3x-2)^6}{(3x-2)^4} $$ Simplifying the powers: $$ (3x-2)^{6-4} = (3x-2)^2 $$

5. Simplifying Expression 5: Start with $$ \frac{2n - 3}{3 - 2n} $$ Factor out -1 from the denominator: $$ \frac{2n - 3}{-(2n - 3)} = -1 $$

6. Simplifying Expression 6: Start with $$ \frac{15y^2(y + 3)}{20y^3(y + 3)} $$ Cancel the common term $(y + 3)$: $$ \frac{15y^2}{20y^3} $$ GCD is 5: $$ \frac{15 \div 5}{20 \div 5} \cdot \frac{y^2}{y^3} = \frac{3}{4} \cdot \frac{1}{y} = \frac{3}{4y} $$

7. Simplifying Expression 7: Start with $$ \frac{15y(y + 1)}{45y^2(y + 1)} $$ Cancel $(y + 1)$: $$ \frac{15y}{45y^2} $$ GCD is 15: $$ \frac{15 \div 15}{45 \div 15} \cdot \frac{1}{y} = \frac{1}{3y} $$

8. Simplifying Expression 8: Start with $$ \frac{24x^3(4 - x)}{18x(4 - x)} $$ Cancel $(4 - x)$: $$ \frac{24x^3}{18x} $$ GCD is 6: $$ \frac{24 \div 6}{18 \div 6} \cdot x^{3-1} = \frac{4}{3} \cdot x^2 = \frac{4x^2}{3} $$

9. Simplifying Expression 9: Start with $$ \frac{6x^3(2 - x)}{12x(2 - x)} $$ Cancel $(2 - x)$: $$ \frac{6x^3}{12x} $$ GCD is 6: $$ \frac{6 \div 6}{12 \div 6} \cdot x^{3-1} = \frac{1}{2} \cdot x