simplify the following algebraic expressions: 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)/... simplify the following algebraic expressions: 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-x-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. Problem 1: Simplifying the fraction

To simplify $\frac{8x^2}{12x^4}$, we can factor out the greatest common divisor (GCD) from the numerator and the denominator.

• The GCD of $8$ and $12$ is $4$.
• We can also cancel $x^2$ from the numerator and $x^4$:

$$ \frac{8x^2}{12x^4} = \frac{8}{12} \cdot \frac{x^2}{x^4} = \frac{2}{3} \cdot \frac{1}{x^2} = \frac{2}{3x^2} $$

2. Problem 2: Simplifying the fraction

For $\frac{24x^2y}{32xy^3}$, we again find the GCD.

• The GCD of $24$ and $32$ is $8$.
• Cancel the common factors $x$ and $y$:

$$ \frac{24x^2y}{32xy^3} = \frac{24}{32} \cdot \frac{x^2}{x} \cdot \frac{1}{y^2} = \frac{3}{4} \cdot x \cdot \frac{1}{y^2} = \frac{3x}{4y^2} $$

3. Problem 3: Simplifying the fraction

For $\frac{(x+1)^3}{(x+1)^5}$:

• We can cancel the common factor of $(x+1)^3$:

$$ \frac{(x+1)^3}{(x+1)^5} = \frac{1}{(x+1)^{2}} $$

4. Problem 4: Simplifying the fraction

For $\frac{(3x-2)^6}{(3x-2)^4}$:

• Again, we cancel the common factor:

$$ \frac{(3x-2)^6}{(3x-2)^4} = (3x-2)^{2} $$

5. Problem 5: Simplifying the fraction

For $\frac{2n-3}{3-2n}$:

• Note that $3-2n = -(2n-3)$, so:

$$ \frac{2n-3}{3-2n} = \frac{2n-3}{-(2n-3)} = -1 $$

6. Problem 6: Simplifying the fraction

For $\frac{15y^2(y+3)}{20y^3(y+3)}$:

• Cancel the common factor $(y + 3)$:

$$ \frac{15y^2}{20y^3} = \frac{15}{20} \cdot \frac{1}{y} = \frac{3}{4y} $$

7. Problem 7: Simplifying the fraction

For $\frac{15y(y+1)}{45y^2(y+1)}$:

• Cancel the common factor $(y + 1)$:

$$ \frac{15y}{45y^2} = \frac{15}{45} \cdot \frac{1}{y} = \frac{1}{3y} $$

8. Problem 8: Simplifying the fraction

For $\frac{24x^3(4-x)}{18x(4-x)}$:

• Cancel the common factor $(4-x)$:

$$ \frac{24x^3}{18x} = \frac{24}{18} \cdot x^{2} = \frac{4}{3}x^{2} $$

9. Problem 9: Simplifying the fraction

For $\frac{6x^3(2-x)}{12x(2-x)}$:

• Cancel $(2-x)$:

$$ \frac{6x^3}{12x} = \frac{6}{12} \cdot x^{2} = \frac{1}{2}x^{2} $$

10. Problem 10: Simplifying the fraction

For $\frac{12x^3(5-2x)}{18(2x-5)}$:

• Note that $2x-5 = -(5-2x)$:

$$ \frac{12x^3(5-2x)}{18(2x-5)} = \frac{12x^3(5-2x)}{-18(5-2x)} = -\frac{12}{18}x^3 = -\frac{2}{3