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.