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