Question

Find the cube root by prime factorization of 10648.

Answer 1

Hint: Here, we have been asked to find the cube root of 10648 by the method of prime factorization. For this, we will first prime factorize 10648. Hence, we will write 10648 as a product of its prime factors. Then we will write 10648 as a product of prime factors with different powers. Then, we will take the cube root of the then obtained equation. On one side, we will have $ \sqrt[3]{10648} $ and on the other side of the equal to sign, there will be its value, i.e. the cube root of 10648. Hence, we will get our answer.

Complete step by step answer:
Here, we have been asked to find the cube root of 10648 by the method of prime factorization.
For this, we will first have to prime factorize 10648. It is done as follows:
\[\begin{align}
  & 2\ \left| \!{\underline {\,
  10648 \,}} \right. \\
 & 2\ \left| \!{\underline {\,
  5324 \,}} \right. \\
 & 2\ \left| \!{\underline {\,
  2662 \,}} \right. \\
 & 11\left| \!{\underline {\,
  1331 \,}} \right. \\
 & 11\left| \!{\underline {\,
  121 \,}} \right. \\
 & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & \ 1\ \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Hence, we can write 10648 as:
 $ \begin{align}
  & 10648=2\times 2\times 2\times 11\times 11\times 11 \\
 & \Rightarrow 10648={{2}^{3}}\times {{11}^{3}} \\
\end{align} $
Now, we have to find its cube root.
For this, let us take cube root on both sides of the obtained equation, i.e. of the equation $ 10648={{2}^{3}}\times {{11}^{3}} $
Thus, taking cube root of the obtained equation on both sides, we get:
 $ \begin{align}
  & 10648={{2}^{3}}\times {{11}^{3}} \\
 & \Rightarrow \sqrt[3]{10648}=\sqrt[3]{{{2}^{3}}\times {{11}^{3}}} \\
 & \Rightarrow \sqrt[3]{10648}=2\times 11 \\
 & \therefore \sqrt[3]{10648}=22 \\
\end{align} $
Thus, we can see that the cube root of 10648 is 22.

Thus, our required answer is 22.

Note:
 Always remember that whenever we have to find any type of root of a number by the method of prime factorization, it can be done by making groups of the same numbers. This is explained as follows:
Here, we have found a cube root. So for finding the cube root of any number, we can group 3 the same numbers together. Here, we can form a group of three 2s and a group of three 11s. Then we will take one group as the same number of which it is a group of. Hence, here we will take the groups of 2s as 2 only and the group of 11s as 11 only. Thus, the cube root will be $ 2\times 11=22 $ .
We can do this for all kinds of roots. If we have to find the square root, we can form a group of 2 same numbers. If we have to find the 4th root, we can form the group of 4 same numbers and so on.