Question

Find the cube root of $9261$.

Answer 1

Hint: To find the cube root of the given number, we will first do the prime factorization of the given number to find the prime factors. Then, we will form the group of three same prime factors. To find the cube root of a number, we count the prime factor that appears thrice in the prime factorization of the number only once.

Complete answer:
So, we will find out the prime factors of the given number $9261$ using the prime factorization method. In the prime factorization method, we break down the original number as the multiplication of its constituent prime factors.
So, we get,
\[\begin{align}
  & 3\left| \!{\underline {\,
  9261 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3087 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1029\,}} \right. \\
 & 7\left| \!{\underline {\,
  343\,}} \right. \\
 & 7\left| \!{\underline {\,
  49\,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
& 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, we get the prime factors of $9261$ as: $3,3,3,7,7,7$ because $9261 = 3 \times 3 \times 3 \times 7 \times 7 \times 7$.
Now, we can see that $3$ and $7$ occur three times in the prime factorization of the number $9261$.
So, we get the cube root of the $9261$ as follows:
$\sqrt[3]{{9261}} = \sqrt[3]{{3 \times 3 \times 3 \times 7 \times 7 \times 7}}$
Taking the number that appear thrice in the prime factorisation of $9261$ outside the bracket, we get,
$ \Rightarrow \sqrt[3]{{9261}} = 3 \times 7$
Simplifying further,
$ \Rightarrow \sqrt[3]{{9261}} = 21$
So, we get the cube root of the number $9261$ as $21$.

Note:
We can also find the cube root of the number using the properties of logarithms and antilogarithms.
Let the cube root of 9261 be x.
$ \Rightarrow x = \sqrt[3]{{9261}}$
We can write the root as raised to $\dfrac{1}{3}$ also.
$ \Rightarrow x = {\left( {9261} \right)^{\dfrac{1}{3}}}$- - - - - - - - - - (1)
Now, to find the cube root of a number using the log method, introduce log on both sides of the equation. Therefore, equation (1) becomes
$ \Rightarrow \log x = \log {\left( {9261} \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( {9261} \right)$- - - - - - - - - (3)
Now, the value of \[\log 9261 = 3.9666578842\]. Therefore, above equation becomes
$ \Rightarrow \log x = \dfrac{1}{3}\left( {3.9666578842} \right)$
$ \Rightarrow \log x = 1.322219294733333$
Now, we need the value of x. So, take antilog on both sides, we get
$\Rightarrow x = anti\log \left( {1.322219294733333} \right)$
$\Rightarrow x = 21$.
Hence, the cube root of 9261 is 21.