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Find the cube root of a given number through estimation: 2197

Answer
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Hint: Start with finding the unit digit of the given number by using the property that says that the unit digit of the product of numbers is the same as the unit digit of the product of the unit digits of the numbers. After finding the unit digit, find the tens multiple between which the required number lies, i.e. 103<2197<203 .

Complete step-by-step answer:
Here in this problem, we are given a positive integer 2197 . We need to find out the cube root of this number through estimation without performing a long calculation.
As we know that the unit digit of the product of two numbers is the same as the unit digit of the product of the unit digits of the two numbers, i.e.
Considering two numbers abcd and wxyz where ‘a’, ‘b’, ‘c’, ‘d’, ‘w’, ‘x’, ‘y’ and ‘z’ are the digits of these two given numbers.
Unit digit of (abcd×wxyz) = Unit digit of (d×z)
If we consider the given number to be the cube of a number ‘m’, then 2197=m3=m×m×m
Then from the above relation, we can say:
Unit digit of (2197) = Unit digit of (m×m×m) = Unit digit of((Unit digit of (m))3)
As we know that the only single-digit number which has 7 as the unit digit of its cube is 3
3×3×3=33=27 Unit digit of (27)=7
Therefore, we can conclude that the unit digit of the required number ‘m’ is 3
Now we can find the number between which the given cube number will lie.
As we know 103=10×10×10=1000 and 203=20×20×20=23×1000=8000
But the given number lies between 1000 and 8000
1000<2197<8000103<21973<20310<m<20
Thus, we came up to the conclusion that the unit digit of the cube root of 2197 is 3 and the cube root of 2197 lies between 10 and 20 .
Therefore, the required cube root of the required number can only be 13 .
21973=13 .

Note: In questions like this, knowing the nearest cubes of the number or finding the cubes of multiples of ten always plays an important role. The cubes or squares of multiples of tens can be easily found by simply multiplying the non-zero together then placing the same number of zeroes from the right. For example, 10×20×30=1×2×3×1000=6×1000=6000 .

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