Question

Find the cube root of the following numbers through estimation \[?\]
(i) \[512\]
(ii) \[2197\]
(iii) \[3375\]
(iv) \[5832\]

Answer 1

Hint: A cube root of a number \[x\] is a number\[y\]such that \[y{}^3 = x\] all non-zero numbers, have exactly one cube root.
Steps to find the cube root by situation:
>To find the cube root by estimation, we need to make a group \[3\] of digit starting from right.
>From the unit digit of groups, we will be estimating that the cube will lie in what interval. Hence, we will be estimating the root.

Complete step-by-step solution:
(i) Cube root of \[512\] :-
We can see that we have only one group.
Let \[0\] be the part of second group
\[\sqrt[3]{{0\,\,512}}\]
If any number is ending with the \[2\] then the cube roots unit digit will be ending in\[8\].
Now \[0\] comes between,
\[0{}^3 \leqslant 0 \leqslant 1{}^3\]
i.e. cube root \[08\] is i.e. \[8\].

(ii) \[\sqrt[3]{{2197}}\]
 We can see that we have two group i.e.
\[\sqrt[3]{{2\,\,197}}\]
If any number is ending in\[7\] , then its cube root's digit must be ending in\[3\].
Now, lies between,
\[1{}^3 \leqslant 2 \leqslant 2{}^3\]
The minimum number is\[1\], Hence the cube root of \[2197\] is\[17\].\[1\] from the second group and \[7\] from the 1st group.

(iii) \[\sqrt[3]{{3375}}\]
 We can see that we have two groups i.e.
\[\sqrt[3]{{3\,\,375}}\]
If any number is ending with \[5\] , then its cube root’s Digit must be ending in \[5\] only .
Now \[3\] lies between,
\[1{}^3 \leqslant 3 \leqslant 2{}^3\]
The minimum number is\[1\] , Hence the cube root of \[3375\] is\[15\].\[1\] from the second group and \[5\] from the first group.

(iv) \[\sqrt[3]{{5832}}\]
We can see that we have two groups
\[\sqrt[3]{{5\,\,\,832}}\]
 If any number is ending with\[2\], Then It’s cube root unit digit must be ending with \[8\].
Now, \[5\] lies between \[1{}^3 \leqslant 5 \leqslant 2{}^3\] i.e. the minimum number is\[1\].
Hence, the cube root of \[5832\] is\[18\].

Note:When we are finding the cube root of any number we have to remember that from the second group we must see that the number lies between whose cubes and then pick the minimum number from it.