Question

How many perfect cubes you can find from 1 to 100? How many from -100 to 100?

Answer 1

Hint: Here, to find the number of perfect cubes between \[1\] to \[100\]calculate the cube of the numbers between \[1\] to \[100\] one by one and counts the number of perfect cubes which are lesser than \[100\] and we get the total number of cubes between \[1\] to \[100\]. And for calculating the number of perfect cubes between \[-100\] to \[100\], calculate the cube of a number between \[-100\] to \[0\] one by one and counts the number of perfect cubes which lies between \[-100\] to \[0\]and then add the total number of cubes between \[1\] to \[100\] as calculated earlier to get the total number of perfect cubes between \[-100\] to \[100\].

Complete step by step answer:
Find the perfect cubes of numbers that lies between \[1\] to \[100\] one at a time
By cube expression,
Solving cube of 1 and 2 we get,
\[{{1}^{3}}=1\times 1\times 1=1\]
\[{{2}^{3}}=2\times 2\times 2=8\]
Solving cube of 3 and 4
\[{{3}^{3}}=3\times 3\times 3=27\]
\[{{4}^{3}}=4\times 4\times 4=64\]
Similarly solving for cube of 5 we get,
\[{{5}^{3}}=5\times 5\times 5=125\text{ }\]
And so on…
As we can see, that the cube of the number \[5\] is greater than \[100\], so it will not be counted.
And the same as with the number greater than \[5\] as their cube will also be greater than \[100\].
This means the cube of the number \[1,2,3\text{ and }4\]lies between \[1\] to\[100\].
So, the number of perfect cubes between \[1\]to\[100\]is\[4\].
Find the perfect cubesof numbers that lies in between \[-100\] to\[0\]one by one:
By cube expression,
\[{{0}^{3}}=0\times 0\times 0\Rightarrow 0\text{ }\]
Finding negative cubes of 1,2,3,4 and 5
\[-{{1}^{3}}=-1\times -1\times -1\Rightarrow -1\]
\[-{{2}^{3}}=-2\times -2\times -2\Rightarrow -8\]
\[-{{3}^{3}}=-3\times -3\times -3\Rightarrow -27\]
\[-{{4}^{3}}=-4\times -4\times -4\Rightarrow -64\]
\[-{{5}^{3}}=-5\times -5\times -5\Rightarrow -125\]
And so on…
As we can see, that the cube of the number \[-5\] does not lie in between \[0\] to \[-100\].
so it will not be counted.
And same as with the number greater than \[-5\].
As their cube will also do not lie in between \[0\] to \[-100\].
This means the cube of the number \[0,-1,-2,-3\text{ }\ \text{and }-4\] lies between \[-100\] to \[0\].
So, the number of perfect cubes between \[-100\] to\[0\]is\[5\].
The total number of cubes lies between \[-100\]to\[100\]:
The perfect cubes of numbers that lies between \[1\] to\[100\]\[+\]the perfect cubes of numbers that lies in between \[0\] to\[-100\]
\[ =4+5 \]
\[ =9\]

Therefore, the total number of cubes lies between \[-100\] to \[100\] is \[9\].

Note:
Note that the cube of 1 is always one and when we multiply two same numbers with negative sign becomes a positive sign multiplying with negative sign becomes negative.
For example-\[-{{1}^{3}}=-1\times -1\times -1\Rightarrow -1\]
The cube of a number \[n\] is its third power, that is, the result of multiplying three instances of \[n\] together.The expression for cube is\[n\times n\times n\].