Question

A rationalising factor of $2^{1/3}+2^{-1/3}$ is

• A. $2^{2/3}-1+2^{-2/3}$
• B. $2^{1/3}-1+2^{-1/3}$
• C. $2^{1/3}+1+2^{-1/3}$
• D. $2^{2/3}+1+2^{-2/3}$

Answer 1

The correct option is A $2^{2/3}-1+2^{-2/3}$

Given,

To find the rationalising factor of $2^{\frac{1}{3}}+2^{-\frac{1}{3}}$

formula, $x^3+y^3=(x+y)(x^2-xy+y^2)$

here, $x=2^{\frac{1}{3}},y=2^{-\frac{1}{3}}$

Therefore, rationalising factor of $2^{\frac{1}{3}}+2^{-\frac{1}{3}}$ is

$=(2^{\frac{1}{3}}{)}^2-2^{\frac{1}{3}}\times 2^{-\frac{1}{3}}+(2^{-\frac{1}{3}}{)}^2$

$=2^{\frac{2}{3}}-1+2^{-\frac{2}{3}}$

verification:
$(2^{\frac{1}{3}}+2^{-\frac{1}{3}})$$(2^{\frac{2}{3}}-1+2^{-\frac{2}{3}})$

$=2-2^{\frac{1}{3}}+2^{\frac{-1}{3}}+2^{\frac{1}{3}}-2^{\frac{-1}{3}}+2^{-1}$

$=2+2^{-1}=\frac{5}{2}$