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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.
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.
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