Question

Is $4096$ a cube number? If yes, find its cube root.

Answer 1

Hint: First of all we need to know what cube number is. Then, we check that $4096$ is a cube number or not. If yes then, we find the cube root of $4096$ by using the prime factorization method. In the prime factorization method we take the triplet of the same number and multiply them to get the cube root of the number.

Complete step-by-step answer:
We have been given a number $4096$.
We have to check $4096$ a cube number. If yes, then we have to find the cube root of $4096$.
Now, we know that we can write $4096$ as $16\times 16\times 16$, it means $4096$ is a cube of $16$ because if a number is multiplied by itself for three times it gives a cube of that number.
Now, we will find the cube root of $4096$ by using the prime factorization method.
In the prime factorization method we will divide the given number by any prime number because we know that a prime number has only two factors i.e. one and the number itself. Now, we know that $2$ is the smallest prime number so let us start dividing $4096$ by $2$.
Now, we have
\[\begin{align}
  & 2\left| \!{\underline {\,
  4096 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2048 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1024 \,}} \right. \\
 & 2\left| \!{\underline {\,
  512 \,}} \right. \\
 & 2\left| \!{\underline {\,
  256 \,}} \right. \\
 & 2\left| \!{\underline {\,
  128 \,}} \right. \\
 & 2\left| \!{\underline {\,
  64 \,}} \right. \\
 & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}\]
Now we have factors of $4096$ as $2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$
Now, we have to make triplets of the factors and take the number only once from the triplet, we have
 $4096=\underbrace{2\times 2\times 2}_{{}}\times \underbrace{2\times 2\times 2}_{{}}\times \underbrace{2\times 2\times 2}_{{}}\times \underbrace{2\times 2\times 2}_{{}}$
$4096=2\times 2\times 2\times 2$
Now, we will multiply the factors to get the cube root of the given number, we have
$4096=16$
So, the cube root of $4096$ is $16$.

Note: In this question we use prime factorization method to find the cube root of the given number. The point to be noted is that using only prime numbers to divide the given number, using any other number gives the wrong answer. Alternatively one can use a long division method.