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;
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;
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