Question

State, with reason, which of the following are surds and which are not:
(i)$\sqrt[3]{{25}}$
(ii)$\sqrt[3]{{40}}$

Answer 1

Hint – In order to solve this problem we need to understand that surd is a number that can't be simplified to remove a square root (or cube root etc.). Simplifying the given number you will know which is surd and which is not.

Complete Step-by-Step solution:
(i) $\sqrt[3]{{25}}$ it is the cube root of 25.
If we factorize 25 then we get 25 = 5x5
And $\sqrt[3]{{25}}$=$\sqrt[3]{{5 \times 5}}$
Therefore we cannot express $\sqrt[3]{{25}}$ free of root so it’s surds.
(ii)$\sqrt[3]{{40}}$ it is the cube root of 40.
If we factorize 25 then we get 40 = 2x2x2x5
And $\sqrt[3]{{40}}$=$\sqrt[3]{{2 \times 2 \times 2 \times 5}}$
Therefore we cannot express $\sqrt[3]{{40}}$ free of root so it’s also surds.

Note – In this problem you need to know about surds. When we can't simplify a number to remove a square root (or cube root etc.) then it is a surd. Example: $\sqrt 2 $ (square root of 2) can't be simplified further so it is a surd. Example: $\sqrt 4 $ (square root of 4) can be simplified (to 2), so it is not a surd. Knowing this can solve your problem.