Question

The number that should be multiplied with $(\sqrt[3]{381}-\sqrt[3]{39}+1)$ to make it rational is

• A. $\sqrt[3]{39}-1$
• B. $\sqrt[3]{39}-3$
• C. $\sqrt[3]{39}+3$
• D. $\sqrt[3]{39}+1$

Answer 1

The correct option is D $\sqrt[3]{39}+1$

$(\sqrt[3]{381}-\sqrt[3]{39}+1)$
$=9^{\frac{2}{3}}-9^{\frac{1}{3}}+1$
$={\left(9^{\frac{1}{3}}\right)}^2-9^{\frac{1}{3}}+1^2$
Now, $a^2-ab+b^2\left)\right(a+b)=a^3+b^3$
$\therefore [{\left(9^{\frac{1}{3}}\right)}^2-9^{\frac{1}{3}}\times 1+1^2](9^{\frac{1}{3}}+1)={\left(9^{\frac{1}{3}}\right)}^3+1^3$
= 9 + 1 =10
Thus, it should be multiplied with $(9^{\frac{1}{3}}+1)$.
Hence, the correct answer is option (d).