Question

What is the cube root of \[\dfrac{1}{125}\]?

Answer 1

Hint: The symbol of cube root is denoted as \[\left( \sqrt[3]{{}} \right)\]. The cube root can be written in exponential form as \[{{\left( {} \right)}^{\dfrac{1}{3}}}\].It means that the number which produces the original number when cubed (multiplied three times with itself).

Complete step by step solution:
Let us understand the concept of cube root with the help of an example;
Suppose, \[\sqrt[3]{a}\] (Where a is called original number).
Any number such that;
\[\Rightarrow a=b\times b\times b\]
We can write \[\left( b\times b\times b \right)\] in place of (a) within the cube root.
\[\Rightarrow \sqrt[3]{b\times b\times b}=b\]

Given that: To find the cube root of \[\dfrac{1}{125}\].
We can write the it as;
\[\Rightarrow \sqrt[3]{\dfrac{1}{125}}\] ………………………… (i)
Equation (i) can be written as;
\[\Rightarrow \dfrac{\sqrt[3]{1}}{\sqrt[3]{125}}\]………………………… (ii)
We know that;
\[1\times 1\times 1=1\] and \[5\times 5\times 5=5\]
The numerator \[\sqrt[3]{1}\]will be equal to 1.
And, denominator \[\sqrt[3]{125}\] will be equal to 5.
Now, equation (ii) can be written as;
\[\dfrac{\sqrt[3]{1}}{\sqrt[3]{125}}=\dfrac{\sqrt[3]{1\times 1\times 1}}{\sqrt[3]{5\times 5\times 5}}=\dfrac{1}{5}\] ………………… (iii)
On division of 1 by 5, we get;
\[\Rightarrow \dfrac{1}{5}=(0.2)\]

Note: If the cube root is to be taken out of a fraction then separate the cube root in numerator and denominator and then find the cube root of numerator and denominator separately. And, if the cube root is to be taken out of a decimal then convert decimal into fraction and solve as we do for a fraction.