Cube and Cube Root
Cube and cube roots are inverse operations when compared to each other. In general, if a number \(X\) is multiplied three times by itself to get \(Y,\) then cube of \(X\) is denoted by \(X^3\) which is equal to \(Y\). On the other hand, cube root of \(Y\) is denoted by \(\sqrt[3]{Y}\) which is equal to \(X.\)
Mathematically, \(X^3=Y\) and \(\sqrt[3]{Y}=X.\)
Cube Root of 9261
Cube root of 9261 is written as \(\sqrt[3]{9261}\). Let us assume that the cube root of 9261 is \(x,\) which is \(\sqrt[3]{9261}=x\).
There are two methods to find out the cube root of a number:
- Prime factorization method
- Estimation method
We will find out the cube root of 9261 using both methods.
Cube Root of 9261 by Prime Factorization Method
Step 1: Find out the prime factors of the given number. Here, we have to find out the prime factors of 9261.
From the table,
9261 = 3 x 3 x 3 x 7 x 7 x 7
Step 2: Group three identical factors in one group.
Step 3: Take one factor from each group.
3 x 7
Step 4: Multiply the taken factors to get the cube root of the given number.
\(\sqrt[3]{9261}=21.\)
So, the cube root of 9261 is 21.
Cube Root of 9261 by Estimation Method
Before applying the estimation method, we must first be familiar with the cube of unit digit numbers.
Step 1: Group the number in two. Take units, tens and hundreds digits to make the first group and second as the rest. Take the first group which is 261 and look at the unit digit of the number. Here, the unit digit of 261 is 1.
Step 2: Check on the table that, which cube has 1 as the unit digit.
From the table, \(1^3=1\), therefore the cube root of 9261 will have 1 at its unit place.
Step 3: Second group is 9, so 9 will be our reference number. From the table, 9 lies between the cubes of 2 and 3.
\(2^3=8\)
\(3^3=27\)
Step 4: Take the number whose cube is nearest to 9. Here, 8 is nearest to 9 so the first number of cube root will be 2.
Step 5: Write the cube root.
\(\sqrt[3]{9261}=21.\)
Solved Examples
Example 1: Find the cube root of 3375.
Solution:
Prime factorization of 3375 is-
3375 = 3 x 3 x 3 x 5 x 5 x 5
\(\sqrt[3]{3375}=3\times5=15\)
So, the cube root of 3375 is 15.
Example 2: Find the cube root of 32768 by estimation method.
Solution:
Group the number 32768 in two, 768 as first and 32 as second.
Unit digit of 768 is 8. So, the cube root of 32768 will have unit digit 2.
The second group is 32 and it lies between the cube of 3 and 4.
Cube of 3 is nearer to 32, so the first number will be 3.
\(\sqrt[3]{32768}=32,\)
Hence, the cube root of 32768 is 32.
Example 3: The volume of a cubical container is 1728 cubic feet. Find the edge length of the container.
Solution:
Let the edge length of the cubical container be \(a\) feet.
The volume of cubical container formula is given by \(a^3\) cubic feet.
The volume of the container is given as 1728 cubic feet.
So, \(a^3=1728\)
\(a=\sqrt[3]{1728}\) feet.
Now, we have to find the cube root of 1728.
1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
\(\sqrt[3]{1728}=2\times2\times3\)
\(\sqrt[3]{1728}=12\)
\(a=12\)
Therefore, the edge length of the cubical container is 12 feet.
The numbers obtained by multiplying a number three times are called perfect cubes. For example two multiplied three times by itself gives eight as the product. Hence, eight is a perfect cube.
There are two methods to find the cube root of any number:
- Prime factorization method
- Estimation method
Yes, the cube of every negative number is negative. For example, negative three multiplied thrice with itself, gives a negative cube, which is negative twenty seven.
The cube of zero is zero.