Theory:

Let us expand some of the cubic terms using their identities.
1. \((2x+3y)^3\)
 
Let us use the identity, \((a+b)^3\)\(=\) \(a^3+3a^2b+3ab^2+b^3\).
 
Comparing \((2x+3y)^3\) with \((a+b)^3\), we have \(a=2x\) and \(b=3y\).
 
Substitute the values in the formula.
 
\((2x+3y)^3\) \(=\) \((2x)^3\)\(+\) \(3(2x)^2(3y)\) \(+\) \(3(2x)(3y)^2\) \(+\) \((3y)^3\)
 
\((2x+3y)^3\) \(=\) \(8x^3\)\(+\) \((3 \times 4 \times 3)x^2y\) \(+\) \((3\times 2\times 9)xy^2\)\(+\) \(27y^3\)
 
\(=\) \(8x^3 + 36x^2y + 54xy^2 + 27y^3\)
 
 
2. \((5x-7y)^3\)
 
Let us use the identity, \((a-b)^3\)\(=\)\(a^3-3a^2b+3ab^2-b^3\).
 
Comparing \((5x-7y)^3\) with \((a-b)^3\), we have \(a=5x\) and \(b=7y\).
 
Substitute the values in the formula.
 
\((5x-7y)^3\) \(=\) \((5x)^3\)\(-\) \(3(5x)^2(7y)\) \(+\) \(3(5x)(7y)^2\) \(-\) \((7y)^3\)
 
\((5x-7y)^3\) \(=\) \(125x^3\)\(-\) \((3\times 25\times 7)x^2y\) \(+\) \((3\times 5 \times 49)xy^2\)\(-\) \(343y^3\)
 
\((5x-7y)^3\) \(=\) \(125x^3\) \(-\) \(525x^2y\) \(+\) \(735xy^2\) \(-\) \(343y^3\)
Example:
Look for the following cases where we used the identities.
 
1. Expand \((y-5)^3\) using the identity.
 
The above expression is of the form \((a-b)^3\).
 
We have the identity, \((a-b)^3\)\(=\)\(a^3-3a^2b+3ab^2-b^3\).
 
Substitute \(a = y\) and \(b = 5\) in the formula.
 
y53=y33(y)2(5)+3(y)(5)253
 
y53=y315y2+75y125
 
 
2. Evaluate \(103^3\) using the identity.
 
Rewrite \(103^3\) as \((100+3)^3\).
 
The above expression is of the form \((a+b)^3\).
 
We have the identity, \((a+b)^3\) \(=\) \(a^3+3a^2b+3ab^2+b^3\)
 
Substitute \(a =100\) and \(b = 3\) in the formula.
 
100+32=1003+3(100)2(3)+3(100)(3)2+33
 
=1000000+(3×10000×3)+(3×100×9)+27
 
=1000000+90000+2700+27
 
=1092727