Question

If in $\sqrt{3}+\sqrt[3]{35}$, x = $\sqrt{3}$ and y = $\sqrt[3]{35}$, then its rationalising factor is :

• A. $x+y$
• B. $x-y$
• C. $x^5+x^{4}y+x^3{y}^2+x^2{y}^3+x{y}^4+y^5$
• D. $x^5-x^{4}y+x^3{y}^2-x^2{y}^3+x{y}^4-y^5$

Answer 1

Answer: (D)

$x^5-x^{4}y+x^3{y}^2-x^2{y}^3+x{y}^4-y^5$

If the given surd is of the form $\sqrt[p]{pa}+\sqrt[q]{qb}$
Let $n$ be the LCM of $p$ and $q$.
If $n$ is even then the rationalising factor is $x^{n-1}-x^{n-2}y+x^{n-3}{y}^2-.....-y^{n-1}$
So, here $p=\frac{1}{2},q=\frac{1}{3}$
$n=\text{LCM of}2,3=6$ (even)
Hence, the rationalising factor is $x^5-x^{4}y+x^3{y}^2-x^2{y}^3+x{y}^4-y^5$