Multiply Rational Expressions
Simplify the rational expressions:
- $\require{cancel}\dfrac{x}{x^2}\times \dfrac{x}{x^2}= \dfrac{\overset{1}{\cancel{x}}}{\underset{x}{\cancel{x^2}}}\times \dfrac{\overset{1}{\cancel{x}}}{\underset{x}{\cancel{x^2}}}= \mathbf{\dfrac{1}{x^2}}$
- $\dfrac{x}{3x}\times \dfrac{2}{x^2}=\dfrac{2x}{3x^3}=\mathbf{\dfrac{2}{3x^2}}$
- $\dfrac{x+1}{x^2}\times \dfrac{3x}{x+1}=\dfrac{\overset{1}{\cancel{(x+1)}}}{x^2}\times \dfrac{3x}{\underset{1}{\cancel{(x+1)}}}=\dfrac{3x}{x^2}=\mathbf{\dfrac{3}{x}} $
- $\dfrac{9x}{x^2+6x+9}\times \dfrac{x^2}{3x^2+12}=\dfrac{\overset{3x}{\cancel{9x}}}{(x+3)(x+3)}\times \dfrac{x^2}{\cancel{3}(x^2+4)}=\mathbf{\dfrac{3x^3}{(x+3)^2(x^2+4)}}$
- $\dfrac{3x+3}{3x^3+6x}\times \dfrac{3x}{x+1}=\dfrac{3\cancel{x+1}}{\cancel{3x}(x^2+2)}\times \dfrac{\cancel{3x}}{\cancel{x+1}}=\mathbf{\dfrac{3}{x^2+2}}$
- $\dfrac{x^2-x-12}{x^2+4x+4}\times \dfrac{x+2}{x-4}=\dfrac{\cancel{(x-4)}(x+3)}{\cancel{(x+2)}(x+2)}\times \dfrac{\cancel{x+2}}{\cancel{x-4}}=\mathbf{\dfrac{x+3}{x+2}}$
- $\dfrac{x^3}{x^2+4x-45}\times \dfrac{x^2+9x}{x^2}=\dfrac{\overset{x}{\cancel{x^3}}}{(x-5)\cancel{(x+9)}}\times \dfrac{x\cancel{(x+9)}}{\cancel{x^2}}=\mathbf{\dfrac{x^2}{x-5}}$
- $\dfrac{x^3-3x^2-40x}{6x}\times \dfrac{3x+12}{x^2-8x}=\dfrac{\cancel{x}(x+5)\cancel{(x-8)}}{\underset{2}{\cancel{6}}\cancel{x}}\times \dfrac{\cancel{3}(x+4)}{x\cancel{(x-8)}}=\mathbf{\dfrac{(x+5)(x+4)}{2x}}$