### 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.

${\left(y-5\right)}^{3}={y}^{3}-3{\left(y\right)}^{2}\left(5\right)+3\left(y\right){\left(5\right)}^{2}-{5}^{3}$

${\left(y-5\right)}^{3}={y}^{3}-15{y}^{2}+75y-125$

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.

${\left(100+3\right)}^{2}={100}^{3}+3{\left(100\right)}^{2}\left(3\right)+3\left(100\right){\left(3\right)}^{2}+{3}^{3}$

$=1000000+\left(3×10000×3\right)+\left(3×100×9\right)+27$

$=1000000+90000+2700+27$

$=1092727$