In this chapter you will learn formula for cube of difference and along with the solved problems at the end.
Cube of Difference definition
The formula for cube of difference is given as;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3ab.( a-b)}
The formula can also be written as;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
The cube of difference of two terms is given by difference of cube of individual terms and addition of \mathtt{-3a^{2} b\ \&\ 3ab^{2}}
Proof of cube of difference formula
The expression given is \mathtt{( a-b)^{3}}
Rewriting the expression;
\mathtt{\Longrightarrow \ ( a-b)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( a-b) \ ( a-b)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( a-b)\left( a^{2} -2ab+b^{2}\right)}
Multiplying the expression;
\mathtt{\Longrightarrow \ a^{3} -2a^{2} b+ab^{2} -a^{2} b+2ab^{2} -b^{3}}\\\ \\ \mathtt{\Longrightarrow \ a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
Hence, we get the formula;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
Proof of cube of difference formula
Let the given expression is \mathtt{( 5-2)^{3}}
Finding value using simple calculation
\mathtt{\Longrightarrow \ ( 5-2)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 3^{3}}\\\ \\ \mathtt{\Longrightarrow \ 27}
Hence, 27 is the value of given expression.
Now let’s find value using the formula;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
Putting the values;
\mathtt{\Longrightarrow \ ( 5-2)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 5^{3} -\ 2^{3} -3( 5)^{2}( 2) \ +3( 5)( 2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 125-8-150+60}\\\ \\ \mathtt{\Longrightarrow \ 27}
The value of given expression is 27.
In both the above methods we got the same value, hence the formula is valid.
Cube of Difference – Solved Problems
Example 01
Expand \mathtt{( 5x-3)^{3}}
Solution
The expression is in form of \mathtt{( a-b)^{3}}
We will use the formula;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
Where;
a = 5x
b = 3
Putting the values we get;
\mathtt{\Longrightarrow \ ( 5x-3)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 5x)^{3} -\ 3^{3} -3( 5x)^{2}( 3) \ +3( 5x)( 3)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 125x^{3} -27-225x^{2} +\ 135x}\\\ \\ \mathtt{\Longrightarrow \ 125x^{3} -225x^{2} +135x-27}
Hence, the above expression is expanded form of given question.
Example 02
Expand \mathtt{\left( 2x-y^{2}\right)^{3}}
Solution
The expression is in form of \mathtt{( a-b)^{3}}
We will use the formula;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
Where;
a = 2x
b = \mathtt{y^{2}}
Putting the values;
\mathtt{\Longrightarrow \ \left( 2x-y^{2}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 2x)^{3} - \left( y^{2}\right)^{3} -3( 2x)^{2}\left( y^{2}\right) \ +3( 2x)\left( y^{2}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 8x^{3} -y^{6} -6x^{2} y^{2} +\ 6xy^{4}}
Hence, the above expression is expanded form of given question.
Example 03
Find the value of \mathtt{27^{3}} using cube of difference formula.
Solution
The number can be written as;
\mathtt{( 27)^{3} \Longrightarrow ( 30-3)^{3}}
\mathtt{( 30-3)^{3}} is in form of expression \mathtt{( a-b)^{3}}
Where;
a = 30
b = 3
We will use the formula;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
Putting the values;
\mathtt{\Longrightarrow ( 30-3)^{3} \ }\\\ \\ \mathtt{\Longrightarrow \ ( 30)^{3} -\ ( 3)^{3} -3( 30)^{2}( 3) \ +3( 30)( 3)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 27000-27-8100+\ 810}\\\ \\ \mathtt{\Longrightarrow \ 19683}
Hence, the value of given expression is 19683.
Example 04
Find the value of \mathtt{( 99)^{3}} using cube of difference formula.
Solution
The number can be written as;
\mathtt{( 99)^{3} \ \Longrightarrow ( 100-1)^{3} \ }
\mathtt{( 100-1)^{3}} is in form of expression \mathtt{( a-b)^{3}}
Where;
a = 100
b = 1
We will use the formula;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3a^{2} b\ +3ab^{2}}
Putting the values;
\mathtt{\Longrightarrow ( 100-1)^{3} \ }\\\ \\ \mathtt{\Longrightarrow \ ( 100)^{3} -\ ( 1)^{3} -3( 100)^{2}( 1) \ +3( 100)( 1)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 1000000-1-30000+300}\\\ \\ \mathtt{\Longrightarrow \ 970299}
Hence, 970299 is the value of given expression.
Example 05
Find the value of \mathtt{x^{3} -64y^{3}}
If x – 4y = 5 and x . y = 12
Solution
It’s given that, x – 4y = 5
Taking cube on both sides;
\mathtt{( x-4y)^{3} =5^{3}}
\mathtt{( x-4y)^{3}} is in form of \mathtt{( a-b)^{3}}
Where;
a = x
b = 4y
We will apply the below formula;
\mathtt{( a-b)^{3} =a^{3} -\ b^{3} -3ab( a-b)}
Using the formula, we get;
\mathtt{x^{3} -64y^{3} -3.x.4y( x-4y) \ =\ 5^{3}}\\\ \\ \mathtt{x^{3} -64y^{3} -12xy( x-4y) \ =\ 5^{3}}\\\ \\ \mathtt{x^{3} -64y^{3} -12.12( 5) \ =125}\\\ \\ \mathtt{x^{3} -64y^{3} -720=125}\\\ \\ \mathtt{x^{3} -64y^{3} \ =\ 845}
Hence, 845 is the value of required expression.
Next chapter : Sum of cube formula