Question

Find the cube root of \[\dfrac{864}{1372}\]?

Answer 1

Hint: From the given question, we have to find the cube root of \[\dfrac{864}{1372}\] . To find the cube root of the \[\dfrac{864}{1372}\] firstly we have to find the prime factorization of the numbers which are present in the numerator and the denominator they are \[864\] and \[1372\] . Prime factorization means finding which prime numbers multiply together to get the original number.

Complete step by step solution:
First, we have to find the prime factorization of \[864\]
Now divide the \[864\] with the \[2\] because \[864\] is an even number and \[2\] is the least prime number which can divide the \[864\] .
On dividing we get,
\[\Rightarrow \dfrac{864}{2}=432\]
Now divide the \[432\] with the \[2\] because \[432\] is an even number and \[2\] is the least prime number which can divide the \[432\] .
On dividing we get,
\[\Rightarrow \dfrac{432}{2}=216\]
 Now divide the \[216\] with the \[2\] because \[216\] is an even number and \[2\] is the least prime number which can divide the \[216\] .
On dividing we get,
\[\Rightarrow \dfrac{216}{2}=108\]
 Now divide the \[108\] with the \[2\] because \[108\] is an even number and \[2\] is the least prime number which can divide the \[108\] .
On dividing we get,
\[\Rightarrow \dfrac{108}{2}=54\]
 Now divide the \[54\] with the \[2\] because \[54\] is an even number and \[2\] is the least prime number which can divide the \[54\] .
On dividing we get,
\[\Rightarrow \dfrac{54}{2}=27\]
 Now divide the \[27\] with the \[3\] because \[27\] is an odd number and the \[3\] is the least prime number which can divide the \[27 \].
On dividing we get,
\[\Rightarrow \dfrac{27}{3}=9\]
 Now divide the \[9\] with the \[3\] because \[9\] is an odd number and the \[3\] is the least prime number which can divide the \[9\].
On dividing we get,
\[\Rightarrow \dfrac{9}{3}=3\]
Now divide the \[3\] with the \[3\] , on dividing we get the remainder zero.
So, the number \[864\] in its prime factors is written as,
\[\Rightarrow 864={{2}^{5}}\times {{3}^{3}}\]
We have to find the cube root. To eliminate the cube root power \[3\] should be there in prime factors.
 So, the \[864\] can be written as, $\Rightarrow 864={{2}^{2}}\times {{6}^{3}}$
Similar way we have to write the prime factorization of \[1372\],
So, the number \[1372\] in its prime factors is written as,
\[\Rightarrow 1372={{7}^{3}}\times {{2}^{2}}\]
Therefore, the cube root of \[\dfrac{864}{1372}\] is $\Rightarrow \sqrt[3]{\dfrac{864}{1372}}=\sqrt[3]{\dfrac{{{6}^{3}}\times {{2}^{2}}}{{{7}^{3}}\times {{2}^{2}}}}=\dfrac{6}{7}$

Note: students should know the method of prime factorization and students should be careful while doing the calculations. The prime factorisation can also be found using below process.
\[\Rightarrow 2\left| \!{\underline {\,
  864 \,}} \right. \]
\[\Rightarrow 2\left| \!{\underline {\,
  432 \,}} \right. \]
\[\Rightarrow 2\left| \!{\underline {\,
  216 \,}} \right. \]
\[\Rightarrow 2\left| \!{\underline {\,
  108 \,}} \right. \]
\[\Rightarrow 2\left| \!{\underline {\,
  54 \,}} \right. \]
\[\Rightarrow 3\left| \!{\underline {\,
  27 \,}} \right. \]
\[\Rightarrow 3\left| \!{\underline {\,
  9 \,}} \right. \]
\[\Rightarrow 3\left| \!{\underline {\,
  3 \,}} \right. \]
\[\Rightarrow 1\] . We can also do this process and find factors and can find the cube root of a given question.