Question

simplify the following rational algebraic expressions. 1. $\frac{9m}{21n}$ 2. $\frac{6p^{2}}{2p - 4p^{3}}$ 3. $\frac{3x + 3y}{x^{2}-y^{2}}$ 4. $\frac{10x^{2}-20x}{5x^{2}-15x}$

Explanation:

Step1: Simplify $\frac{9m}{21n}$

Find the greatest - common divisor of 9 and 21. GCD(9, 21)=3. Then $\frac{9m}{21n}=\frac{3\times3m}{3\times7n}=\frac{3m}{7n}$.

Step2: Simplify $\frac{6p^{2}}{2p - 4p^{3}}$

Factor out the greatest - common factor from the denominator. $2p-4p^{3}=2p(1 - 2p^{2})$. Then $\frac{6p^{2}}{2p - 4p^{3}}=\frac{2p\times3p}{2p(1 - 2p^{2})}=\frac{3p}{1 - 2p^{2}}$.

Step3: Simplify $\frac{3x + 3y}{x^{2}-y^{2}}$

Factor out 3 from the numerator: $3x + 3y=3(x + y)$. Factor the denominator using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$, so $x^{2}-y^{2}=(x + y)(x - y)$. Then $\frac{3x + 3y}{x^{2}-y^{2}}=\frac{3(x + y)}{(x + y)(x - y)}=\frac{3}{x - y}$.

Step4: Simplify $\frac{10x^{2}-20x}{5x^{2}-15x}$

Factor out the greatest - common factor from the numerator and denominator. The numerator $10x^{2}-20x = 10x(x - 2)=2\times5x(x - 2)$, and the denominator $5x^{2}-15x=5x(x - 3)$. Then $\frac{10x^{2}-20x}{5x^{2}-15x}=\frac{2\times5x(x - 2)}{5x(x - 3)}=\frac{2(x - 2)}{x - 3}$.

Answer:

1. $\frac{3m}{7n}$
2. $\frac{3p}{1 - 2p^{2}}$
3. $\frac{3}{x - y}$
4. $\frac{2(x - 2)}{x - 3}$