Question

What is the cube root of $96$?

Answer 1

Hint: To find the cube root of $96$, we are going to use the method of log. First of all, let the cube root of $96$ be x. Now, cube root can also be denoted by raising the number by $\dfrac{1}{3}$. So, we will get the equation as $x = {\left( {96} \right)^{\dfrac{1}{3}}}$. Now, take log on both sides and then simplify RHS. After that, to find the value of x, take antilog on both sides and we will get our answer.

Complete step-by-step solution:
In this question, we are supposed to find the cube root of 12.5.
We can easily find this using a calculator, but we are going to see a method to find the cube root of any number without using a calculator.
For this, we are going to use the log method.
First of all, the cube root of a number means the number which when multiplied three times will give the original number. Cube root of a number is denoted by $\sqrt[3]{{}}$.
Let the cube root of $96$ be $x$.
$ \Rightarrow x = \sqrt[3]{{96}}$
We can also write cube roots as raised to $\dfrac{1}{3}$.
$ \Rightarrow x = {\left( {96} \right)^{\dfrac{1}{3}}}$ - - - - - - (1)
Now, to find the cube root of a number using log method, introduce log on both sides of the equation.
Therefore, equation (1) becomes
$ \Rightarrow \log x = \log {\left( {96} \right)^{\dfrac{1}{3}}}$ - - - - - - - - (2)
Now, we have the property $\log {a^b} = b\log a$. Therefore, equation (2) becomes
$ \Rightarrow \log x = \dfrac{1}{3}\log \left( {96} \right)$- - - - - - - (3)
Now, the value of $\log 96 = 1.98227$. Therefore, equation (3) becomes
$ \Rightarrow \log x = \dfrac{1}{3}\left( {1.98227} \right)$
$ \Rightarrow \log x = 0.660757$
Now, we need the value of x. So, take antilog on both sides, we get
$
   \Rightarrow x = anti\log \left( {0.660757} \right) \\
   \Rightarrow x = {\text{4}}{\text{.578856}} \\
 $
Hence, the cube root of $96$ is \[{\text{4}}{\text{.578856}}\].

Note: Here, we can cross check our answer by multiplying \[{\text{4}}{\text{.578856}}\] three times.
$ \to {\text{4}}{\text{.578856}} \times {\text{4}}{\text{.578856}} \times {\text{4}}{\text{.578856}} = 95.999939$
Hence, our answer is correct.