# Cube of sum

In this chapter we will learn cube of sum formula and will also solve questions related to above concept.

## Cube of Sum definition

The formula for sum of cube is given by;

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

The cube of sum of two number is given by sum of cube of individual numbers and addition of \mathtt{3a^{2} b\ \&\ 3ab^{2}}

You can also write this formula as;

\mathtt{( a+b)^{3} =a^{3} +\ b^{3} +3ab.( a+b)}

## Derivation of cube of sum 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 sum formula

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

Finding value using simple calculation;

\mathtt{\Longrightarrow \ ( 2+5)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 7)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 343}

Hence, 343 is the value of given expression.

Now let’s find value of expression using the formula;

\mathtt{( a+b)^{3} =a^{3} +\ b^{3} +3a^{2} b+3ab^{2}} \\\ \\
\mathtt{\Longrightarrow 2^{3} +\ 5^{3} +3\ ( 2)^{2} 5+3( 2)( 5)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 8\ +\ 125+60+150}\\\ \\ \mathtt{\Longrightarrow \ 343}

The value of given expression is 343.

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

## Cube of Sum -Solved Problems

Example 01
Expand \mathtt{( 2x+3)^{3}}

Solution
The expression is in the form \mathtt{( a+b)^{3}}

We will use the formula;

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

Putting the values we get;
\mathtt{( 2x+3)^{3}}

\mathtt{\Longrightarrow ( 2x)^{3} +\ 3^{3} +3\ ( 2x)^{2} 3+3( 2x)( 3)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 8x^{3} +27+36x^{2} +54x}\\\ \\ \mathtt{\Longrightarrow \ 8x^{3} +36x^{2} +54x+27}

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

Example 02
Expand \mathtt{( 6x-2)^{3}}

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

Where;
a = 6x
b = -2

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

Putting the values;

\mathtt{\Longrightarrow \ ( 6x-2)^{3}}\\\ \\ \mathtt{\Longrightarrow ( 6x)^{3} +( -2)^{3} +3\ ( 6x)^{2}( -2) +3( 6x)( -2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 216x^{3} -8-216x^{2} +72x}\\\ \\ \mathtt{\Longrightarrow \ 216x^{3} -216x^{2} +72x-8}

The above expression is the expanded form of given problem.

Example 03
Expand \mathtt{\left( -4-y^{2}\right)^{3}}

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

Where;
a = -4
b = \mathtt{-y^{2}}

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

Putting the values;

\mathtt{\Longrightarrow \ \left( -4-y^{2}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow ( -4)^{3} +\left( -y^{2}\right)^{3} +3\ ( -4)^{2}\left( -y^{2}\right) +3( -4)\left( -y^{2}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -64-y^{6} -48y^{2} -12y^{4}}\\\ \\ \mathtt{\Longrightarrow -y^{6} -12y^{4} -48y^{2} -64}

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

Example 04
Find the value of \mathtt{( 107)^{3}} using cube of sum formula.

Solution
The number can be written as;
\mathtt{( 107)^{3} \ \Longrightarrow \ ( 100+7)^{3}}

\mathtt{( 100+7)^{3}} is in form of expression \mathtt{( a+b)^{3}}

Where;
a = 100
b = 7

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

Putting the values;

\mathtt{\Longrightarrow \ ( 107)^{3} \ \Longrightarrow \ ( 100+7)^{3}}\\\ \\ \mathtt{\Longrightarrow ( 100)^{3} +( 7)^{3} +3\ ( 100)^{2}( 7) +3( 100)( 7)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 10000+343+210000+14700}\\\ \\ \mathtt{\Longrightarrow 1225043}

Hence, 1225043 is the value of given expression.

Example 05
Find the value of \mathtt{( 19)^{3}} using sum of cube formula.

Solution
The number can be written as;
\mathtt{( 19)^{3} \ \Longrightarrow \ ( 20-1\ )^{3}}

The number is in form of expression \mathtt{( a+b)^{3}}

Where;
a = 20
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 \ ( 19)^{3} \ }\\\ \\ \mathtt{\Longrightarrow \ ( 20-1\ )^{3}}\\\ \\ \mathtt{\Longrightarrow ( 20)^{3} +( -1)^{3} +3\ ( 20)^{2}( -1) +3( 20)( -1)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 8000-1-1200+60}\\\ \\ \mathtt{\Longrightarrow 6859}

Hence, 6859 is the value of given number.

Example 06
Find the value of \mathtt{8x^{3} +\ 125y^{3}} .
If 2x + 5y = 3 and x . y=2

Solution
We know that 2x + 5y = 3

Taking cube on both sides;

\mathtt{( 2x+5y)^{3} \ =\ 3^{3}}\\\ \\ \mathtt{( 2x+5y)^{3} =27}

Expanding \mathtt{2x+5y)^{3}} using formula \mathtt{( a+b)^{3}}

We know that;
\mathtt{( a+b)^{3} =a^{3} +\ b^{3} +3ab.( a+b)}

Putting the values, we get;

\mathtt{( 2x)^{3} +( 5y)^{3} +3( 2x)( 5y) .( 2x+5y) \ =\ 27}\\\ \\ \mathtt{\ 8x^{3} +125y^{3} +30xy( 2x+5y) \ =\ 27}

It’s given that;
If 2x + 5y = 3 and x . y=2

Putting the values in above equation we get;

\mathtt{8x^{3} +125y^{3} \ +30\times 2\times 3=27}\\\ \\ \mathtt{8x^{3} +125y^{3} \ +180\ =\ 27}\\\ \\ \mathtt{8x^{3} +125y^{3} \ =\ 27-180}\\\ \\ \mathtt{8x^{3} +125y^{3} \ =\ -153}

Hence, value of \mathtt{8x^{3} +\ 125y^{3}} is -153

Next chapter : cube of difference formula