# Sum of Cube

In this chapter, we learn sum of cubes formula and will also solve some questions related to the concept.

## Sum of Cube formula

The formula for sum of cube is given as;

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

Memorize the formula, as it would help us to solve different algebra questions in math exam.

### Proof of sum of cube formula

Let the given expression is \mathtt{5^{3} +2^{3}}

Finding value using simple calculation;

\mathtt{\Longrightarrow \ 5^{3} +2^{3}}\\\ \\ \mathtt{\Longrightarrow \ 125\ +8}\\\ \\ \mathtt{\Longrightarrow \ 133}

Hence, 133 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} +2^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 5+2)\left( 5^{2} -5.2+2^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 7)( 25-10+4)}\\\ \\ \mathtt{\Longrightarrow \ 133}

The value of given expression is 133.

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

## Sum of Cubes – Solved Problems

Example 01
Expand \mathtt{216\ x^{3} +125\ y^{3}}

Solution
The expression can be written as;

\mathtt{216\ x^{3} +125\ y^{3} \Longrightarrow \ ( 6x)^{3} +( 5y)^{3}}

The expression \mathtt{( 6x)^{3} +( 5y)^{3}} is 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 \ ( 6x)^{3} +( 5y)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 6x+5y)\left(( 6x)^{2} -6x.5y+( 5y)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 6x+5y)\left( 36x^{2} -30xy+25y^{2}\right)}

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

Example 02
Expand \mathtt{343\ x^{3} +729\ y^{3}}

Solution
The expression can be written as;
\mathtt{343\ x^{3} +729\ y^{3} \Longrightarrow \ ( 7x)^{3} +( 9y)^{3}}

The expression \mathtt{( 7x)^{3} +( 9y)^{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 \ ( 7x)^{3} +( 9y)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 7x+9y)\left(( 7x)^{2} -7x.9y+( 9y)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 7x+9y)\left( 49x^{2} -63xy+81y^{2}\right)}

The above expression is expanded form of given problem.

Example 3
Expand \mathtt{5x^{3} +\ 625y^{3}}

Solution
The expression can be simplified as;

\mathtt{\Longrightarrow \ 5x^{3} +\ 625y^{3} \ }\\\ \\ \mathtt{\Longrightarrow \ 5\ \left( x^{3} +125y^{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ 5\ \left( x^{3} +\ ( 5y)^{3}\right)}

The expression \mathtt{x^{3} +\ ( 5y)^{3}} is in the form of \mathtt{a^{3} +b^{3}}

Applying the sum of cube formula we get;

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

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

Example 04
Expand \mathtt{\ 256x^{3} +\ 4y^{3} \ }

Solution
Simplifying the given expression;

\mathtt{\Longrightarrow \ 256x^{3} +\ 4y^{3} \ }\\\ \\ \mathtt{\Longrightarrow 4\left( 64x^{3} +y^{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ 4\left(( 4x)^{3} +\ y^{3}\right)}

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

Solving the expression using sum of cube formula;

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

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

Example 05
Find the value of \mathtt{10^{3} +\ 11^{3}} using sum of cube formula.

Solution
The expression \mathtt{10^{3} +\ 11^{3}} is in the form of \mathtt{a^{3} +b^{3} \ }

Expanding the expression with sum of cube formula, we get;

\mathtt{\Longrightarrow \ 10^{3} +\ 11^{3} \ }\\\ \\ \mathtt{\Longrightarrow ( 10+11)\left(( 10)^{2} -10.11+( 11)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 21)( 100-110+121)}\\\ \\ \mathtt{\Longrightarrow \ 21\ .\ 111}\\\ \\ \mathtt{\Longrightarrow \ 2331}

Hence, 2331 is the value of given expression.

Next chapter : Difference of cube formula