Question

The cube root of \[.000216\] is
A. \[.6\]
B. \[.06\]
C. \[77\]
D. \[87\]

Answer 1

Hint: We write the number given in the statement in the form of a fraction by removing its decimal and dividing it by power of ten which is equal to the number of digits after the decimal place. Then we find the cube root of both numerator and denominator.
* Cube root of any number is that number which when multiplied to itself gives us the cube of the number.
* If a number is in the decimal form and the number of digits after the decimal are m, then we write the number in the form of a fraction by dividing the number from \[{10^m}\].

Complete step-by-step answer:
We are given the number in the statement as \[.000216\]
We write this number in the form of a fraction.
Since, number of digits after the decimal are 6, so we write the numerator as the number itself without the decimal in it and denominator as \[{10^6}\]
\[ \Rightarrow .000216 = \dfrac{{000216}}{{{{10}^6}}}\]
Since the zeroes before the number in the numerator don’t affect the number, we remove them.
\[ \Rightarrow .000216 = \dfrac{{216}}{{{{10}^6}}}\]
Now we write the numerator and denominator in the terms of its factors.
We can write \[216 = 6 \times 6 \times 6 = {6^3}\]
Using law of indices \[{a^{m \times n}} = {({a^m})^n}\] we can write \[{10^{3 \times 2}} = {({10^2})^3}\]
Substituting the values in numerator and denominator.
\[ \Rightarrow .000216 = \dfrac{{{6^3}}}{{{{({{10}^2})}^3}}}\]
Now we take cube root on both sides of the equation
\[ \Rightarrow \sqrt[3]{{.000216}} = \sqrt[3]{{\dfrac{{{6^3}}}{{{{({{10}^2})}^3}}}}}\]
Since we know, cube root of a number can be written as \[\sqrt[3]{a} = {(a)^{\dfrac{1}{3}}}\]
\[ \Rightarrow \sqrt[3]{{.000216}} = {\left( {\dfrac{{{6^3}}}{{{{({{10}^2})}^3}}}} \right)^{\dfrac{1}{3}}}\]
We can write using the law of indices \[{({a^m})^n} = {a^{m \times n}}\]
\[
   \Rightarrow \sqrt[3]{{.000216}} = \left( {\dfrac{{{6^{3 \times \dfrac{1}{3}}}}}{{{{({{10}^2})}^{3 \times \dfrac{1}{3}}}}}} \right) \\
   \Rightarrow \sqrt[3]{{.000216}} = \left( {\dfrac{6}{{({{10}^2})}}} \right) \\
   \Rightarrow \sqrt[3]{{.000216}} = \left( {\dfrac{6}{{100}}} \right) \\
 \]
Now we see that the denominator of the RHS has \[{10^2}\]. So we can convert the fraction into decimal form by placing the decimal after two digits when starting from the right side of the number.
We can write \[\dfrac{6}{{100}} = .06\]
\[ \Rightarrow \sqrt[3]{{.000216}} = .06\]
Thus, cube root of \[.000216\] is \[.06\]

So, the correct answer is “Option B”.

Note: Students many times make mistakes while converting the number from fraction to decimal form because they start putting the decimal value from left side of the number which is wrong, always start from the backside i.e. right side and move to left side of the number.