# How do you use the binomial theorem to expand and simplify the expression (1/x+y)^5?

Dec 19, 2017

${\left(\frac{1}{x} + y\right)}^{5} = \frac{1}{{x}^{5}} + \frac{5 y}{x} ^ 4 + \frac{10 {y}^{2}}{x} ^ 3 + \frac{10 {y}^{3}}{x} ^ 2 + \frac{5 {y}^{4}}{x} + {y}^{5}$

#### Explanation:

${\left(\frac{1}{x} + y\right)}^{5}$

=${\left(\frac{1}{x}\right)}^{5} {y}^{0} \cdot C \left(5 , 0\right) + {\left(\frac{1}{x}\right)}^{4} {y}^{1} \cdot C \left(5 , 1\right) + {\left(\frac{1}{x}\right)}^{3} {y}^{2} \cdot C \left(5 , 2\right) + {\left(\frac{1}{x}\right)}^{2} {y}^{3} \cdot C \left(5 , 3\right) + {\left(\frac{1}{x}\right)}^{1} {y}^{4} \cdot C \left(5 , 4\right) + {\left(\frac{1}{x}\right)}^{0} {y}^{5} \cdot C \left(5 , 5\right)$

=$\frac{1}{{x}^{5}} + \frac{5 y}{x} ^ 4 + \frac{10 {y}^{2}}{x} ^ 3 + \frac{10 {y}^{3}}{x} ^ 2 + \frac{5 {y}^{4}}{x} + {y}^{5}$