Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube?

Last updated date: 14th Jul 2024
Total views: 346.2k
Views today: 10.46k
Verified
346.2k+ views
Hint: We try to form the indices formula for the value 3. This is finding the cube root of 9720. We find the prime factorisation of 9720. Then we take one digit out of the three same number of primes. There will be an odd number of primes remaining in the root which can’t be taken out.

We try to find the value of the algebraic form of $\sqrt[3]{9720}$. This is a cube root form.
We know the theorem of indices ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$. Putting value 3 we get ${{a}^{\dfrac{1}{3}}}=\sqrt[3]{a}$.
\begin{align} & 2\left| \!{\underline {\, 9720 \,}} \right. \\ & 2\left| \!{\underline {\, 4860 \,}} \right. \\ & 2\left| \!{\underline {\, 2430 \,}} \right. \\ & 3\left| \!{\underline {\, 1215 \,}} \right. \\ & 3\left| \!{\underline {\, 405 \,}} \right. \\ & 3\left| \!{\underline {\, 135 \,}} \right. \\ & 3\left| \!{\underline {\, 45 \,}} \right. \\ & 3\left| \!{\underline {\, 15 \,}} \right. \\ & 5\left| \!{\underline {\, 5 \,}} \right. \\ & 1\left| \!{\underline {\, 1 \,}} \right. \\ \end{align}
Therefore, $9720=2\times 2\times 2\times 3\times 3\times 3\times 3\times 3\times 5$.
Therefore, to make it a perfect cube we divide 9720 with the least number of $3\times 3\times 5=45$.
Note: We can also use the variable form where we can take $x=\sqrt[3]{9720}$. But we need to remember that we can’t use the cube on both sides of the equation $x=\sqrt[3]{9720}$ as in that case we are taking two extra values as a root value. Then this linear equation becomes a cubic equation.