Question

How do you simplify the cube root of $40$?

Answer 1

Hint:Here we must know what the cube root of the number means. When we need to find the cube root of the number, we actually need to find the number which when multiplied to itself three times gives the result as the number under the cube root. In this way, we can simplify the cube root of the number.

Complete step by step solution:
Here we are given to simplify the cube root of the number which is $40$
We must know that the cube root of the number is denoted by the symbol $\sqrt[3]{n}$ where $n$ can be any number whose cube root is to be found.
We can write it in the form of its factors. If any factor is occurring three times that number can be taken outside the cube root and written once and other occurring only once or twice will remain in the cube root. The example will make it clearer.
If we need to find the cube root of $16$ we can write it as $\sqrt[3]{{16}}$.
Now we can write it in the form of factors as:
$\sqrt[3]{{(2)(2)(2)(2)}}$
Here we can see that $2$ is occurring four times so we can take out one $2$ for the three $2s$and then one will be left inside. So we can simplify it as:
$2\sqrt[3]{2}$
Now we are given to find the cube root of $40$ which is $\sqrt[3]{{40}}$.
We know that its factors are as:
$
  40 = (2)(20) \\
  20 = (2)(10) \\
  10 = (2)(5) \\
  5 = (5)(1) \\
 $
Hence we can write $40 = (2)(2)(2)(5)$
So we need to find $\sqrt[3]{{(2)(2)(2)(5)}}$
Now we have three $2{\text{s}}$ which can be taken out of the cube root so we will get:
$\sqrt[3]{{40}} = 2\sqrt[3]{5}$.

Note: Here the student must know that if he is asked to find the square root of the number then he needs to write it in the form of factors in the same way but in square root, we can take out of the root those numbers who are in pair which means occur twice in the root.