# Difference of cubes

In this chapter we will learn difference of cubes formula and some solved examples related to the concept.

## Difference of Cubes formula

The formula for difference of cube is given as;

\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

The formula is important. Memorize it to solve algebra related problems.

### Proof of Difference of Cube formula

Let the given expression is \mathtt{5^{3} -3^{3}}

Finding value using simple calculation;

\mathtt{\Longrightarrow \ 5^{3} -3^{3}}\\\ \\ \mathtt{\Longrightarrow \ 125\ -\ 27}\\\ \\ \mathtt{\Longrightarrow \ 98}

Hence, 98 is the value of given expression.

Now let’s find value using the formula;

\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

Putting the values;

\mathtt{\Longrightarrow \ 5^{3} -3^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 5-3)\left( 5^{2} +5.3+3^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ 2.\ ( 25+15+9)}\\\ \\ \mathtt{\Longrightarrow \ 2.\ 49}\\\ \\ \mathtt{\Longrightarrow \ 98\ }

The value of given expression is 98.

In both the methods we got the same value, hence the formula is valid.

## Difference of Cubes – Solved Problems

Example 01
Expand \mathtt{343\ x^{3} -64\ y^{3}}

Solution
The expression can be written as;

\mathtt{\Longrightarrow \ 343\ x^{3} -64\ y^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 7x)^{3} -( 4y)^{3}}

The expression \mathtt{( 7x)^{3} -( 4y)^{3}} is in the form of \mathtt{a^{3} -b^{3}}

We will use the formula;
\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

Putting the values, we get;

\mathtt{\Longrightarrow \ ( 7x)^{3} -( 4y)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 7x-4y)\left(( 7x)^{2} +7x.4y+( 4y)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 7x-4y) .\left( 49x^{2} +28xy+49y^{2}\right)}

Hence, the above expression is the expanded form of given problem.

Example 02
Expand \mathtt{48x^{3} -750y^{3}}

Solution
The expression can be written as:

\mathtt{\Longrightarrow \ 48x^{3} -750y^{3}}\\\ \\ \mathtt{\Longrightarrow \ 6\left( 8x^{3} -125y^{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ 6\left(( 2x)^{3} -( 5y)^{3}\right)}

The given term is in the form of \mathtt{a^{3} -b^{3}}

We will use the formula;
\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

Putting the values;

\mathtt{\Longrightarrow \ 6\left(( 2x)^{3} -( 5y)^{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ 6\ ( 2x-5y)\left(( 2x)^{2} +2x.5y+( 5y)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ 6( 2x-5y) .\left( 4x^{2} +10xy+25y^{2}\right)}

Hence, the above expression is expanded form of given problem.

Example 03
Fine the value of \mathtt{13^{3} -10^{3}} using difference of cube formula.

Solution
The expression \mathtt{13^{3} -10^{3}} is in the form of \mathtt{a^{3} -b^{3}}

We will use the formula;
\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

Putting the values;

\mathtt{\Longrightarrow \ 13^{3} -10^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 13-10)\left(( 13)^{2} +13.10+( 10)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 3) .( 169\ +130+100)}\\\ \\ \mathtt{\Longrightarrow \ 1197}

Hence, 1197 is the value of given expression.

Example 04
Find the value of \mathtt{25^{3} -20^{3}}

Solution
The expression \mathtt{25^{3} -20^{3}} is in the form of \mathtt{a^{3} -b^{3}}

We will use the formula;
\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

Putting the values;

\mathtt{\Longrightarrow \ 25^{3} -20^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 25-20)\left(( 25)^{2} +25.20+( 20)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 5) .( 625\ +500+400)}\\\ \\ \mathtt{\Longrightarrow \ 7625}

Hence, 7625 is the value of given expression.

Example 05
Expand \mathtt{1024x^{3} -16y^{3}}

Solution
The expression can be written as;

\mathtt{\Longrightarrow \ 1024x^{3} -16y^{3}}\\\ \\ \mathtt{\Longrightarrow \ 2\left( 512x^{3} -8y^{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ 2\ \left(( 8x)^{3} -( 2y)^{3}\right)}

The expression is in form of \mathtt{a^{3} -b^{3}}

We will use the formula;
\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

Putting the values;

\mathtt{\Longrightarrow \ 2\ \left(( 8x)^{3} -( 2y)^{3}\right)}\\\ \\ \mathtt{\Longrightarrow 2( 8x-2y)\left(( 8x)^{2} +8x.2y+( 2y)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ 2( 8x-2y) .\left( 16x^{2} \ +16xy+4y^{2}\right)}

Hence, the above expression is the expanded form of given question.