Question

Find the cube root of 6859 by the prime factorization method?

Answer 1

Hint: To find the cube root of 6859, we are going to do the prime factorization of 6859. And then we will take the three values at a time and when we apply the cube root on the complete prime factorization then those three values which we have taken at a time then only one value out of three will come out.

Complete step by step answer:
In the above problem, we are asked to find the cube root of 6859 using the prime factorization method.
So, we are going to do the prime factorization of 6859. We will divide this number by the lowest possible natural number, then whatsoever the quotient will remain, we will see by which number that number will get divisible. In this way, we will divide till we get the quotient as 1.
The number which 6859 is completely divisible is 19 so dividing 6859 by 19 we get,
$\begin{matrix}
   19 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 6859 \\
 & 361 \\
\end{align} \,}} \right. $
Now, by hit and trial method, you will find that 361 is divisible by 19 so dividing 361 by 19 we get,
$\begin{matrix}
   19 \\
   19 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 6859 \\
 & 361 \\
 & 19 \\
\end{align} \,}} \right. $
After that 19 is divisible by 19 and we get,
\[\begin{matrix}
   19 \\
   19 \\
   19 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 6859 \\
 & 361 \\
 & 19 \\
 & 1 \\
\end{align} \,}} \right. \]
From the above, we have found the prime factorization of 6859 as follows:
$6859=19\times 19\times 19$
Now, taking cube root on both the sides of the above equation we get,
${{\left( 6859 \right)}^{\dfrac{1}{3}}}={{\left( 19\times 19\times 19 \right)}^{\dfrac{1}{3}}}$ …………….. (1)
We know the property of the exponents as follows:
${{a}^{x}}\times {{a}^{y}}\times {{a}^{z}}={{a}^{x+y+z}}$
Now, applying the above property in $19\times 19\times 19$ we get,
$19\times 19\times 19={{19}^{3}}$
Using the above relation in eq. (1) we get,
$\begin{align}
  & {{\left( 6859 \right)}^{\dfrac{1}{3}}}={{\left( {{19}^{3}} \right)}^{\dfrac{1}{3}}} \\
 & \Rightarrow {{\left( 6859 \right)}^{\dfrac{1}{3}}}={{\left( 19 \right)}^{3\times \dfrac{1}{3}}} \\
\end{align}$
In the exponent of the R.H.S of the above equation, 3 will get cancelled out from the numerator and the denominator and we get,
$\begin{align}
  & \Rightarrow {{\left( 6859 \right)}^{\dfrac{1}{3}}}={{\left( 19 \right)}^{1}} \\
 & \Rightarrow {{\left( 6859 \right)}^{\dfrac{1}{3}}}=19 \\
\end{align}$

Hence, we have found the cube root of 6859 as 19.

Note: We can check the cube root of 6859 by multiplying 19 by 3 times and on multiplication, if we get the value 6859 then the cube root we have found is correct.
Multiplying 19 three times we get,
$\begin{align}
  & 19\times 19\times 19=361\times 19 \\
 & \Rightarrow 19\times 19\times 19=6859 \\
\end{align}$
As you can see that on multiplication, we are getting the same number 6859 so the cube root we have found out is correct.