Question

Find the cube root of $27\times 64$.

Answer 1

Hint: For solving this question first, we will write $27\times 64$ into multiples of prime numbers and then apply the cube root operator. After that, we will find the value of the cube root of the number .

Complete step-by-step answer:
Given:
We have to find the value of the cube root of $27\times 64$ .
Now, before we proceed we should know that prime numbers are the numbers which are greater than 1 and have only two factors 1 and the number itself.
Now, first, we will write $27\times 64$ into multiples of prime numbers only. Then,
$\begin{align}
  & 27\times 64=3\times 3\times 3\times 2\times 2\times 2\times 2\times 2\times 2 \\
 & \Rightarrow 27\times 64={{3}^{3}}\times {{2}^{6}} \\
\end{align}$
Now, from the above equation, we can find the value of the cube root of the number $27\times 64$ easily. So, we will now find the cube root of $27\times 64$ . Then,
\[\begin{align}
  & 27\times 64={{3}^{3}}\times {{2}^{6}} \\
 & \Rightarrow \sqrt[3]{27\times 64}=\sqrt[3]{{{3}^{3}}\times {{2}^{6}}} \\
 & \Rightarrow \sqrt[3]{27\times 64}=\sqrt[3]{{{3}^{3}}\times {{\left( {{2}^{2}} \right)}^{3}}} \\
 & \Rightarrow \sqrt[3]{27\times 64}=3\times {{2}^{2}} \\
 & \Rightarrow \sqrt[3]{27\times 64}=3\times 4 \\
 & \Rightarrow \sqrt[3]{27\times 64}=12 \\
\end{align}\]
Now, from the result of the above calculation that we did, we can write that the value of the cube root of $27\times 64$ will be equal to 12.

Note: Here, the student should first understand what is asked in the problem. Although the problem is very easy, and while factorising $27\times 64$ we should be careful and not miss any prime number while writing $27\times 64$ into multiples of prime numbers. After that, we should apply the cube root operator and find the value of the cube root.