# Cube of Difference

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