# How to solve:- binomial expansion (1+2x)^1/2

Welcome to my article binomial expansion (1+2x)^1/2. This question is taken from the simplification lesson.
The solution of this question has been explained in a very simple way by a well-known teacher by doing addition, subtraction, and fractions.
For complete information on how to solve this question binomial expansion (1+2x)^1/2, read and understand it carefully till the end.

Let us know how to solve this question binomial expansion (1+2x)^1/2.

First write the question on the page of the notebook.

## binomial expansion (1+2x)^1/2

According to dinomeat series-

latex \displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+\frac{{2x}}{2}+\frac{{\left( {\frac{1}{2}} \right)\left( {\frac{1}{2}-1} \right)}}{{2!}}{{(2x)}^{2}}+\frac{{\left( {\frac{1}{2}} \right)\left( {\frac{1}{2}-1} \right)\left( {\frac{1}{2}-2} \right)}}{{3!}}{{(2x)}^{3}}+\frac{{\left( {\frac{1}{2}} \right)\left( {\frac{1}{2}-1} \right)\left( {\frac{1}{2}-2} \right)\left( {\frac{1}{2}-3} \right)}}{{4!}}{{(2x)}^{4}}+\text{.................................}.

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x+\frac{{\left( {\frac{1}{2}} \right)\left( {-\frac{1}{2}} \right)}}{{2!}}4{{x}^{2}}+\frac{{\left( {\frac{1}{2}} \right)\left( {-\frac{1}{2}} \right)\left( {-\frac{3}{2}} \right)}}{{3!}}8{{x}^{3}}+\frac{{\left( {\frac{1}{2}} \right)\left( {-\frac{1}{2}} \right)\left( {-\frac{3}{2}} \right)\left( {-\frac{5}{2}} \right)}}{{4!}}16{{x}^{4}}+…………….

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{1}{8}4{{x}^{2}}+\frac{3}{{48}}8{{x}^{3}}-\frac{{15}}{{384}}16{{x}^{4}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{{1\times 4{{x}^{2}}}}{8}+\frac{{3\times 8{{x}^{3}}}}{{48}}-\frac{{15\times 16{{x}^{4}}}}{{384}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{{4{{x}^{2}}}}{8}+\frac{{24{{x}^{3}}}}{{48}}-\frac{{15\times 16{{x}^{4}}}}{{384}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{{4{{x}^{2}}}}{8}+\frac{{24{{x}^{3}}}}{{48}}-\frac{{15\times 16{{x}^{4}}}}{{24\times 16}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{4}{8}{{x}^{2}}+\frac{{24}}{{48}}{{x}^{3}}-\frac{{15\times 16}}{{24\times 16}}{{x}^{4}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{{4\times 1}}{{4\times 2}}{{x}^{2}}+\frac{{24\times 1}}{{24\times 2}}{{x}^{3}}-\frac{{16\times 15}}{{16\times 24}}{{x}^{4}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{{4\times 1}}{{4\times 2}}{{x}^{2}}+\frac{{24\times 1}}{{24\times 2}}{{x}^{3}}-\frac{{16\times 15}}{{16\times 24}}{{x}^{4}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{1}{2}{{x}^{2}}+\frac{1}{2}{{x}^{3}}-\frac{{15}}{{24}}{{x}^{4}}+……..]

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{1}{2}{{x}^{2}}+\frac{1}{2}{{x}^{3}}-\frac{{3\times 5}}{{3\times 8}}{{x}^{4}}+……..

\displaystyle {{\left( {1+2x} \right)}^{{\frac{1}{2}}}}=1+x-\frac{1}{2}{{x}^{2}}+\frac{1}{2}{{x}^{3}}-\frac{5}{8}{{x}^{4}}+……..