Question

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

Answer 1

Hint: Perfect cubes are those numbers which are cubes of any integer. Here, instead of finding the perfect cubes, find the numbers whose cubes are below 100. This will make the process a bit easier. And the property of perfect cubes is also applicable for negative numbers. If $x = {y^3}$ then $ - x = {\left( { - y} \right)^3}$. So use this property and imitate the results from positive numbers.

Complete step-by-step solution:
According to the question, we have to determine the number of perfect cubes lying from 1 to 100 and from -100 to 100.
First we’ll consider positive integers only.
We know that the perfect cubes are those numbers which are cubes of integers. In that respect we’ll find the natural numbers whose cubes are less than 100. We’ll start with 1 and move forward.
$
   \Rightarrow {1^3} = 1 \\
   \Rightarrow {2^3} = 8 \\
   \Rightarrow {3^3} = 27 \\
   \Rightarrow {4^3} = 64 \\
   \Rightarrow {5^3} = 125 \\
   \Rightarrow {6^3} = 216
 $
We can see that as we move towards 5 and above the cube roots are more than 100. So we can say that the positive integers whose cubes are 100 or below are 1, 2, 3 and 4 only.
Thus there are 4 perfect cubes from 1 to 100 and they are 1, 8, 27 and 64.
For negative numbers we can directly imitate the results from positive numbers as we know that if $x = {y^3}$ then $ - x = {\left( { - y} \right)^3}$. So we have:
\[
   \Rightarrow {\left( { - 1} \right)^3} = - 1 \\
   \Rightarrow {\left( { - 2} \right)^3} = - 8 \\
   \Rightarrow {\left( { - 3} \right)^3} = - 27 \\
   \Rightarrow {\left( { - 4} \right)^3} = - 64
 \]

Thus there are 4 perfect cubes from -100 to -1. They are -1, -8, -27 and -64.
But between -100 to 100 we also have to include 0 as we know that ${0^3} = 0$.
Therefore together from -100 to 100, we have 4 negative perfect cubes, 4 positive perfect cubes and 0. Hence from -100 to 100, there are 9 perfect cubes.

Note: We can take the cube root of negative numbers and we will get a real number which is also negative. But the square root of a negative number will not give us a real number. This will be an imaginary number. And imaginary numbers are represented by a letter $i$ which is equal to $\sqrt { - 1} $. So we can use them as shown below:
\[
   \Rightarrow \sqrt { - 1} = i \\
   \Rightarrow \sqrt { - 4} = 2\sqrt { - 1} = 2i
 \]