Search any question & find its solution
Question:
Answered & Verified by Expert
Which one of the following is correct? The real number $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$ is:
Options:
Solution:
2075 Upvotes
Verified Answer
The correct answer is:
a rational number but not an integer
The given number
$\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$
can be written as
$=(2+\sqrt{5})^{1 / 3}+(2-\sqrt{5})^{1 / 3}$
$=2^{1 / 3}\left[1+\frac{1}{2} \sqrt{5}\right]^{1 / 3}+2^{1 / 3}\left[1-\frac{1}{2} \sqrt{5}\right]^{1 / 3}$
$=2^{1 / 3}\left[1+\frac{1}{6} \sqrt{5}+\ldots+1-\frac{1}{6} \sqrt{5}+\ldots\right]$
Thus the given number is a rational number but not an integer
$\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$
can be written as
$=(2+\sqrt{5})^{1 / 3}+(2-\sqrt{5})^{1 / 3}$
$=2^{1 / 3}\left[1+\frac{1}{2} \sqrt{5}\right]^{1 / 3}+2^{1 / 3}\left[1-\frac{1}{2} \sqrt{5}\right]^{1 / 3}$
$=2^{1 / 3}\left[1+\frac{1}{6} \sqrt{5}+\ldots+1-\frac{1}{6} \sqrt{5}+\ldots\right]$
Thus the given number is a rational number but not an integer
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.