Karen Morrison and Nick Hamshaw Solutions for Exercise 14: Exercise 14.14

Write the following as a single fraction in its lowest terms:

$\frac{5}{2xy}-\frac{4}{3xy}$

$\frac{2}{x}+x$

Writing above as a single fraction in its lowest terms, we get $\frac{2+x^k}{x}$.

Write the value of $k$.

$\frac{x+1}{2}+\frac{2}{x+1}$

Writing above as a single fraction in its lowest terms, we get $\frac{x^2+2x+k}{2\left(x+1\right)}$.

Write the value of $k$.

$\frac{3(x^2-1)}{7y}-\frac{2(x^2-1)}{9y^2}$

Writing above as a single fraction in its lowest terms, we get $\frac{(x^2-1)(27y-k)}{\left(x+1\right)\left(x+4\right)}$.

Write the value of $k$.

$\frac{1}{x^2}-\frac{x}{2y}$

Writing above as a single fraction in its lowest terms, we get $\frac{2y-x^3}{k{x}^{2}y}$.

Write the value of $k$.

$\frac{x+1}{3z^2}-\frac{y+z}{12xy}$

Writing above as a single fraction in its lowest terms, we get $\frac{4x^{2}y+4xy-y{z}^2-z^3}{kxy{z}^2}$.

Write the value of $k$.

Write the following as a single fraction in its lowest terms:

$\frac{1}{\left(x+2\right)}-\frac{1}{\left(x+3\right)\left(x+2\right)}$

If the following expression can be simplified in a single fraction in its lowest terms as $\frac{2}{x+1}-\frac{2}{x^2+3x+2}=\frac{k}{x+k}$, then find $k$.