Question

$\text{Let}x=(\sqrt{50}+7{)}^{\frac{1}{3}}-(\sqrt{50}-7{)}^{\frac{1}{3}}.$ 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

The correct option is A $x=2$

$x=(\sqrt{50}+7{)}^{\frac{1}{3}}-(\sqrt{50}-7{)}^{\frac{1}{3}}$
Let $t=(\sqrt{50}+7{)}^{\frac{1}{3}}\Rightarrow \frac{1}{t}=(\sqrt{50}-7{)}^{\frac{1}{3}}$
$x=t-\frac{1}{t}{x}^3=t^3-\frac{1}{t^3}-3(t-\frac{1}{t})$
$x^3+3x=\sqrt{50}+7-\sqrt{50}+7=14$
$x^3+3x-14=0\Rightarrow (x-2)(x^2+2x+7)=0$
$\Rightarrow x=2$