Question

The number of rational terms in the expansion of $(1+\sqrt{2}+\sqrt[3]{33}{)}^6$ is

• A. $6$
• B. $7$
• C. $3$
• D. $8$

Answer 1

The correct option is B $7$

$(1+2^{1/2}+3^{1/3}{)}^6$
$=\left[\right(1+2^{1/2}+3^{1/3}{)}^3{]}^2$
$=\left[\right(1+(2^{1/2}+3^{1/3}{)}^3+3(2^{1/2}+3^{1/3}{)}^2+3(2^{1/2}+3^{1/3}){]}^2$
$=1+(2^{1/2}+3^{1/3}{)}^6+9(2^{1/2}+3^{1/3}{)}^4+9(2^{1/2}+3^{1/3}{)}^2+2(2^{1/2}+3^{1/3}{)}^3+6(2^{1/2}+3^{1/3}{)}^5+18(2^{1/2}+3^{1/3}{)}^3+6(2^{1/2}+3^{1/3})$
Hence the rational terms are
$1,24,4,12,6,9,8$
Hence, there are total $7$ rational terms