How do you multiply \[{{\left( 7x-5y \right)}^{2}}?\]

Hint: We are asked to multiply \[{{\left( 7x-5y \right)}^{2}}\] and we will start by observing how many terms are there in the given item and then we will try to find a suitable identity to solve this. We will also multiply using the basic approach and we will also solve using algebraic identity to get the different ways possible to solve the same problem.

Complete step-by-step solution:
We are given \[{{\left( 7x-5y \right)}^{2}}\] and as power is 2 it means that we have to multiply (7x – 5y) with itself. There are various ways to solve the problem. We will start by the basic approach in which we multiply (7x – 5y) with itself by the multiplication method in which the single term of the first bracket is multiplied by each term of the second bracket. So, we get,
\[\left( 7x-5y \right)\left( 7x-5y \right)=7x\left( 7x-5y \right)-5y\left( 7x-5y \right)\]
Opening the brackets we will see
\[\Rightarrow \left( 7x-5y \right)\left( 7x-5y \right)=7x\times 7x-7x\times 5y-5y\times 7x-5y\times \left( -5y \right)\]
On simplifying, we get,
\[\Rightarrow \left( 7x-5y \right)\left( 7x-5y \right)=49{{x}^{2}}-35yx-35xy+25{{y}^{2}}\]
Adding the like terms, we get,
\[\Rightarrow \left( 7x-5y \right)\left( 7x-5y \right)=49{{x}^{2}}-70xy+25{{y}^{2}}\]
Hence, we get,
\[\Rightarrow \left( 7x-5y \right)\left( 7x-5y \right)=49{{x}^{2}}-70xy+25{{y}^{2}}\]
Now we will work on the other way in which we will use Binomial identity as in our problem we have two terms only. So, we can use Binomial identities here. We will use \[{{\left( A-B \right)}^{2}}={{A}^{2}}-2AB+{{B}^{2}}.\] So, on using \[{{\left( A-B \right)}^{2}}\] on \[{{\left( 7x-5y \right)}^{2}}\] we will consider A = 7x and B = 5y. So, we get,
\[{{\left( 7x-5y \right)}^{2}}={{\left( 7x \right)}^{2}}-2\times 7x\times 5x+{{\left( 5y \right)}^{2}}\]
On simplifying, we get,
\[\Rightarrow {{\left( 7x-5y \right)}^{2}}=49{{x}^{2}}-70xy+25{{y}^{2}}\]
Therefore, we get,
\[{{\left( 7x-5y \right)}^{2}}=49{{x}^{2}}-70xy+25{{y}^{2}}\]

Note: Remember that when we multiply the variables, their power gets added up which is \[x\times x={{x}^{2}}.\] So, like \[x\times x=2x\] cannot be made. Also, remember that the like terms can only be added. So, adding \[9{{x}^{2}}-1y=8{{x}^{2}}y\] is a mistake as \[9{{x}^{2}}\] and 1y are not like terms at all. Also, we need to be careful that \[{{\left( 2x \right)}^{2}}\ne 4x\] as the square is applied over both 2 as well as x. Also, remember when two negative terms are added, the term gets added by the same sign. For example – 6x – 6x = – 12x.