Question

Calculate cube root: 8000

Answer 1

Hint: We will first start by prime factorization the number. Then we will use the fact that for finding the cube root a pair factor inside the cube root becomes one outside. So, using this we will find the cube root of the given number.

Complete step-by-step answer:
Now, we have been given that the number is 8000.
Now we know that the prime factorisation of a number is the product of all the prime factors of the number for example prime factorisation of 50 is $50=2\times 5\times 5$
Now we can see that the prime factorisation of 50 has two 5 and one 2.
Now similarly we can see that the prime factorization of 8000 is
$8000=2\times 2\times 2\times 2\times 2\times 2\times 5\times 5\times 5$
Now we can see that the prime factorization of 8000 has six 2 and three 5 because if we divide by 8000 by 2 for exactly six times by 2 then we will left with a number that is indivisible by 2 and in prime factorisation we have to find the prime number which divides 8000 similarly if we divide the number 8000 with 5 three times then we will left with a number that will be further indivisible by 5.
Now, we know that, $a\times a\times a..........n\ times={{a}^{n}}$, therefore the prime factorization of 8000 is
$8000={{2}^{3}}\times {{2}^{3}}\times {{5}^{3}}........\left( 1 \right)$
Now, we know that the cube root of a number whose prime factorization is given can be found by bundling the factors each in a group of 3 and then only one from such group comes outside the cube root. So, we have,
$\begin{align}
  & \sqrt[3]{8000}=\sqrt[2]{{{2}^{3}}\times {{2}^{3}}\times {{5}^{3}}} \\
 & =2\times 2\times 5 \\
 & =4\times 5 \\
 & =20 \\
\end{align}$
Hence, the cube root of 8000 is 20.

Note: To solve this question it is important to note that we have prime factored the number and formed a group of 3 of each factor and only one of such groups come out of the cube root because the cube root of a number is a number which when multiplied three times gives the original number.