Question

What is the cube root of $ \dfrac{1}{4} $ ?

Answer 1

Hint: To find the cube root of $ \dfrac{1}{4} $ , we are going to use the method of log. First of all, let the cube root of $ \dfrac{1}{4} $ 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( {\dfrac{1}{4}} \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 answer:
In this question we have to find the cube root of $ \dfrac{1}{4} $ . Now, we can find its cube root using the long division method or using the log method. But the long division method will be a little complicated so 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 gives the original number. Cube root is denoted by $ \sqrt[3]{{}} $ .
Let the cube root of $ \dfrac{1}{4} $ be x.
 $ \Rightarrow x = \sqrt[3]{{\dfrac{1}{4}}} $
We can write the root as raised to $ \dfrac{1}{3} $ also.
 $ \Rightarrow x = {\left( {\dfrac{1}{4}} \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( {\dfrac{1}{4}} \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( {\dfrac{1}{4}} \right) $ - - - - - - - - - (3)
Now, we can write $ \log \dfrac{a}{b} = \log a - \log b $ . Therefore, equation (3) becomes
 $ \Rightarrow \log x = \dfrac{1}{3}\left( {\log 1 - \log 4} \right) $ - - - - - - (4)
Now, we know that $ \log 1 = 0 $ . Therefore equation (4) becomes
 $
   \Rightarrow \log x = \dfrac{1}{3}\left( {0 - \log 4} \right) \\
   \Rightarrow \log x = \dfrac{1}{3}\left( { - \log 4} \right) \\
   \Rightarrow \log x = - \dfrac{1}{3}\left( {\log 4} \right) \;
  $
Now, the value of $ \log 4 = 0.602 $ . Therefore, above equation becomes
 $ \Rightarrow \log x = - \dfrac{1}{3}\left( {0.602} \right) $
 $ \Rightarrow \log x = - 0.200686 $
Now, we need the value of x. So, take antilog on both sides, we get
 $
   \Rightarrow x = anti\log \left( { - 0.200686} \right) \\
   \Rightarrow x = {\text{0}}{\text{.6299702}} \;
  $ .
Hence, the cube root of $ \dfrac{1}{4} $ is $ {\text{0}}{\text{.6299702}} $ .
So, the correct answer is “ $ {\text{0}}{\text{.6299702}} $ ”.

Note: Here, we can verify our answer by multiplying our answer three times.
 $ {\text{0}}{\text{.6299702}} \times {\text{0}}{\text{.6299702}} \times {\text{0}}{\text{.6299702}} = 0.25001 $ .
And, $ \dfrac{1}{4} = 0.25 $ . Hence, our answer is correct. We can find any root of a given number using the log method.