Question

Find the cube root of 0.027.

Answer 1

Hint: Make the decimal representation of the number to fractional representation. Split the numerator and denominator in a way to take the cube root easier.

Complete step-by-step answer:
A cube root of a number x is a number y such that \[{{y}^{3}}=x\].
All non – zero real numbers have exactly one real cube root and a pair of complex conjugate cube roots and all non – zero complex numbers have three distinct complex cube roots.
Given to us, \[\sqrt[3]{0.027}\].
We know the decimal representation of 0.027 can be written as \[\dfrac{27}{1000}\].
\[\therefore \sqrt[3]{0.027}=\sqrt[3]{\dfrac{27}{1000}}\]
Thus we can write it as,
\[\sqrt[3]{\dfrac{3\times 3\times 3}{10\times 10\times 10}}=\dfrac{\sqrt[3]{{{3}^{3}}}}{\sqrt[3]{{{10}^{3}}}}=\dfrac{3}{10}\]
\[\therefore \]Decimal representation of \[\dfrac{3}{10}\]= 0.3.
Thus we got, \[\sqrt[3]{0.027}=0.3\].

Note: Note that, \[\sqrt[3]{{{a}^{3}}}=\sqrt{a\times a\times a}=a\].
Similarly, if it was, \[\sqrt[3]{27}=\sqrt[3]{3\times 3\times 3}=3\].
We can find the cube root of any non – zero real number like this. If given in decimal form convert to fractional form to make it easier.