Question

Simplify$\sqrt[3]{{375}}$.

Answer 1

Hint: For simplifying the terms containing cube root we have to remove all the perfect cubes, if present any inside the cube root. So here we have to factorize the given term and represent it in terms of perfect cubes and then simplify it.

Complete step-by-step solution:
Given
$\sqrt[3]{{375}}..............................................\left( i \right)$
Now we have to change the given term$\sqrt[3]{{375}}$and represent them in terms of perfect cubes such that we can take one of the terms out of the three outside the cube root and thereby simplify it.
Now first we have to factorize $375$.
\[\Rightarrow 375 = 3 \times 5 \times 5 \times 5...........................\left( {ii} \right)\]
On observing (ii) we see that the factorization of $392$gives\[3 \times 5 \times 5 \times 5\].
Now we need to find $\sqrt[3]{{375}}$ which is \[\sqrt[3]{{3 \times 5 \times 5 \times 5}}\].
Now we know that the terms which are present inside three times out of it one of the terms can be taken out of the cube root.
So here we can see that the number \[5\] is present three times inside the cube root, such that we can take one of the numbers which is one $5$ outside the cube root. Thereby after taking $5$ outside the cube root we have only $3$ inside the cube root which is to be kept untouched.
Therefore we can write:
\[
\Rightarrow \sqrt[3]{{375}} = \sqrt[3]{{3 \times 5 \times 5 \times 5}} \\
 \Rightarrow 5 \times \sqrt[3]{3} \\
 \Rightarrow 5\sqrt[3]{3}............................\left( {iii} \right) \\
 \]

Therefore on simplifying $\sqrt[3]{{375}}$we get\[5\sqrt[3]{3}\].

Note: Radical expressions are algebraic expressions which have or contain radicals, and the best way to solve a cube root is to remove all the perfect cubes from inside the cube root if any exists. Also questions similar can be solved in a similar manner which is to factorize the given number and then taking one of the three digits outside the cube root if multiple numbers exist.