Question

Let $x=\left(\sqrt{50}+7\right)^{1 /3}-\left(\sqrt{50}-7\right)^{1 /3}$. Then,

• A. $x=2$
• B. $x=3$
• C. $x$ is a rational number, but not an integer
• D. $x$ is an irrational number

Answer 1

Answer: (A)

Given,
$x=\left(\sqrt{50}+7\right)^{1 /3}-\left(\sqrt{50}-7\right)^{1/ 3}$
On cubing both sides, we get
$x^{3}=\left(\sqrt{50}+7\right)-\left(\sqrt{50}-7\right)-3$
$\left(\sqrt{50}+7\right)^{1 /3} \left(\sqrt{50}-7\right)^{1 /3}$
$\Rightarrow \left[\left(\sqrt{50}+7\right)^{1/ 3}-\left(\sqrt{50}-7\right)^{1/ 3}\right]$
$\Rightarrow x^{3}=14-3\left(50-49\right)^{1 /3} \left(x\right)$
$\Rightarrow x^{3}=14-3x$
$\Rightarrow x^{3}+3x-14=0$
$\Rightarrow \left(x-2\right)\left(x^{2}+2x+7\right)=0$
$\Rightarrow x=2$