In this chapter we will learn about the concept of perfect cubes along with the method to calculate cube root of given number.
What is Perfect Cube ?
A perfect cube is a number formed when we multiply integer thrice by itself.
Hence, when a number is cubed, we get a perfect cube.
For example;
Let’s multiply the number 5 by itself three times.
\mathtt{\Longrightarrow \ 5\times 5\times 5}\\\ \\ \mathtt{\Longrightarrow \ 125}
Here number 125 is a perfect cube since it is formed by multiplying number 3 times.
Example 02
Multiply number 12 by itself three times.
\mathtt{\Longrightarrow \ 12\times 12\times 12}\\\ \\ \mathtt{\Longrightarrow \ 1728}
Here the number 1728 is the perfect cube.
Check if number is perfect cube or not ?
Suppose you are given a number 3375.
To check if the below number is perfect cube or not, follow the below steps;
(i) Find prime factorization given number
(ii) Arrange all the factors with exponent 3
(iii) If all the factors are arranged with power 3 then the given number is a perfect cube.
(iv) Remove all the exponents and multiply the factor to get the cube root.
Let’s take the example of number 3375
(i) Do the prime factorization.
(ii) Write all the factors produced in prime factorization.
\mathtt{3375\ =3^{3} \times 5^{3}}
Note that all the factors have power 3, hence the number 3375 is a perfect cube.
(iii) Remove all the exponent of three to get the cube root.
Cube Root 3375 ⟹ 3 x 5
Cube root 3375 ⟹ 15
Hence, if we multiply the number 15 by itself thrice, we get number 3375.
Example 02
Check if 512 is perfect cube
Solution
Follow the below steps;
(a) Do the prime factorization
(ii) The factorization can be expressed as;
\mathtt{512\ \Longrightarrow \ 2^{3} \times 2^{3} \times 2^{3}}
In the calculation, try to form a pair of three for every given factor.
Factor 2 ⟹ form pair of three
Factor 2 ⟹ form pair of three
Factor 2 ⟹ form pair of three
Since all the factors are arranged with exponent 3, the number 512 is a perfect cube.
(iii) Remove all exponent of 3 to get the cube root
Cube Root 512 ⟹ 2 x 2 x 2
Cube root 512 ⟹ 8
Hence, if we multiply 8 by itself thrice we get 512.
Example 03
Check if number 108 is perfect cube
Solution
Follow the below steps;
(a) Do the prime factorization
The prime factorization is given as;
\mathtt{108\ \Longrightarrow \ 2^{2} \times 3^{3}}
(b) Check if all factors forming pair of three.
Factor 2 ⟹ doesn’t form pair of 3
Factor 3 ⟹ form pair of 3
Since the factor doesn’t form pair of three, the number 108 is not a perfect cube.
Example 04
Check if number 1000 is a perfect cube
Solution
Follow the below steps;
(a) Do the prime factorization
The above prime factorization is expressed as;
\mathtt{1000\ \Longrightarrow \ 2^{3} \times 5^{3}}
(ii) Check if all factor form pair of three
Factor 2 ⟹ form pair of 3
Factor 5 ⟹ form pair of 3
Since all the factors have 3 as an exponent, the number 1000 is a perfect cube.
(iii) Remove the exponent and multiply the number, you will get the cube root.
Cube Root 1000 ⟹ 2 x 5
Cube root 1000 ⟹ 10
Hence, if we multiply number 10 by itself thrice, we get 1000.
Example 05
Check if number 1715 is perfect cube.
Solution
(i) Do the prime factorization
The prime factorization is expressed as;
\mathtt{1715\ \Longrightarrow \ 5\times 7^{3}}
(ii) Check if all factor form pair of three.
Factor 5 ⟹ doesn’t form pair of 3
Factor 7 ⟹ form pair of 3
Since the factor 5 doesn’t contain exponent of three, the number 1715 is not a perfect cube.