Question

How do you find all the real cube roots of $\dfrac{8}{{125}}$

Answer 1

Hint: Here we have to find the real cube roots of the given values. Also, we split the given values by the fraction term and finally we get the required answer.

Complete step-by-step solution:
We have the given number as:
$ \Rightarrow \dfrac{8}{{125}}$
Since we have to find the cube root of the equation, we can write it as:
$ \Rightarrow \sqrt[3]{{\dfrac{8}{{125}}}}$
Now since the number is in the form of a fraction which cannot be reduced anymore, we will split the cube root in the numerator and denominator separately.
Therefore, it can be written as:
$ \Rightarrow \dfrac{{\sqrt[3]{8}}}{{\sqrt[3]{{125}}}}$
Now we will expand both the numerator and denominator, it can be written as:
$ \Rightarrow \dfrac{{\sqrt[3]{{2 \times 2 \times 2}}}}{{\sqrt[3]{{5 \times 5 \times 5}}}}$
Since the numbers are in cube root and they are multiplied thrice we can take out the cube root by writing it only once.
$ \Rightarrow \dfrac{2}{5}$

Therefore,
$ \Rightarrow \sqrt[3]{{\dfrac{8}{{125}}}} = \dfrac{2}{5}$, which is the required answer.

Note: The cube root we have found in this question is the real cube root of the number, the complex roots can be found as: $\dfrac{2}{5}\omega $ and $\dfrac{2}{5}{\omega ^2}$
We know the value of $\omega $ which is the complex root of unity which has value $\omega = \dfrac{{ - 1 + i\sqrt 3 }}{2}$
And the value of ${\omega ^2}$ is ${\omega ^2} = \dfrac{{ - 1 - i\sqrt 3 }}{2}$
On simplifying and getting the roots we get:
$\dfrac{2}{5}\omega = \dfrac{2}{5}\left( {\dfrac{{ - 1 + i\sqrt 3 }}{2}} \right) = \dfrac{{ - 1 + i\sqrt 3 }}{5}$
And the other root is: $\dfrac{2}{5}{\omega ^2} = \dfrac{2}{5}\left( {\dfrac{{ - 1 - i\sqrt 3 }}{2}} \right) = \dfrac{{ - 1 + i\sqrt 3 }}{5}$
It is to be remembered that complex numbers are also called imaginary numbers and it is used when we have to find the square root of a negative number.
The value of $i$ is $\sqrt { - 1} $ and it is used when finding the square root of a negative number.
The application of imaginary or complex numbers is in physics when there is computation in the imaginary form.