Question

The digit at unit place of the cube root of $ 2197 $ is
 $ \begin{align}
  & a)\,3 \\
 & b)\,9 \\
 & c)\,7 \\
 & d)\,6 \\
\end{align} $

Answer 1

Hint: We are given a number $ 2197 $. We have to find the unit place of the cube root of $ 2197 $, we first factorize the $ 2197 $, once we have the prime factor, we then use it to find the cube root. Using the cube root, we will find the unit place of it.
There is another way in which we use option one by one, find their cube, the option whose cube has unit place same as unit place of $ 2197 $ is the required option.

Complete step by step answer:
We are given a number $ 2197 $, we are asked to find the unit place of the cube root of this number.
Mean we have to find which term will be then at the unit place of the cube root of the $ 2197 $.
To find the cube root of any number we multiply that number three times. Similarly, in opposite direction, to find the cube of any number, we make a pair of \[3\] number and report one in place of pair of three to find the cube root. We first factorize and then simplify.
We have $ 2197 $, we will write the prime factorization of $ 2197 $.
We have that $ 2197 $ is nothing but the product of $ 13 $ three times.
That is
 $ 2197=13\times 13\times 13 $
So,
 $ 2197={{\left( 13 \right)}^{3}} $
If we take cube root on both sides we get,
 $ \sqrt[3]{2197}\,=\sqrt[3]{{{\left( 13 \right)}^{3}}}\, $
Now we simplify we get:
 $ \sqrt[3]{2197}\,=\sqrt[3]{{{\left( 13 \right)}^{3}}}\,=\sqrt[3]{13\times 13\times 13} $
Out of cube root, one in place of \[3\] term will come out so we get
 $ \sqrt[3]{2197}\,=13\, $
We see that cube root is $ 13 $ , in which one place is \[3\] and ten places is \[1\]
So, we get in the cube root of the unit $ 2197 $ the unit place is occupied by \[3\].
Hence our required answer is \[3\]
So, option a) \[3\] is correct.


Note:
 Another way to find the solution is to check the following one by one.
We will find each term available in the option, the term whose cube unit place is the same as $ 2197 $ unit place the that will be our answer.
So, for a) \[3\]
\[{{3}^{3}}=3\times 3\times 3=9\times 32=27\]
Unit place is \[7\] also.
So, this, is the correct option.
We can see that
c) \[{{7}^{3}}=343\]
unit place is \[3\] and
d) \[6\]
\[{{6}^{3}}=216\]
Unit place is \[6\]
b) \[9\]
\[{{9}^{3}}=729\]
Unit place is \[9\]
So, none of the option b, c, d have unit place \[7\]. So, option a) \[3\] is correct.