# How do you use the binomial theorem to expand and simplify the expression (x+2y)^4?

${\left(x + 2 y\right)}^{4} = {x}^{4} + 8 {x}^{3} y + 24 {x}^{2} {y}^{2} + 32 x {y}^{3} + 16 {y}^{4}$

#### Explanation:

Using Binomial expansion of ${\left(x + 2 y\right)}^{4}$ as follows

${\left(x + 2 y\right)}^{4}$

${=}^{4} {C}_{0} {x}^{4} {+}^{4} {C}_{1} {x}^{3} \left(2 y\right) {+}^{4} {C}_{2} {x}^{2} {\left(2 y\right)}^{2} {+}^{4} {C}_{3} x {\left(2 y\right)}^{3} {+}^{4} {C}_{4} {\left(2 y\right)}^{4}$

$= {x}^{4} + 4 {x}^{3} \left(2 y\right) + 6 {x}^{2} \left(4 {y}^{2}\right) + 4 x \left(8 {y}^{3}\right) + 16 {y}^{4}$

$= {x}^{4} + 8 {x}^{3} y + 24 {x}^{2} {y}^{2} + 32 x {y}^{3} + 16 {y}^{4}$