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**Hint:**We first express the square as a product of two like terms, that is $\left( y-7 \right)\times \left( y-7 \right)$ . Then we apply distributive property to the expression to get $y\times \left( y-7 \right)-7\times \left( y-7 \right)$ . Having done so, we again have to apply distributive property to get ${{y}^{2}}-7y-7\times \left( y-7 \right)$ which upon simplification gives ${{y}^{2}}-14y+49$ .

**Complete step by step answer:**

The given expression is

${{\left( y-7 \right)}^{2}}$

By finding the product of the above given expression means to evaluate the square of the expression $\left( y-7 \right)$ as is given in the question. Now, the expression ${{\left( y-7 \right)}^{2}}$ can also be written as $\left( y-7 \right)\times \left( y-7 \right)$ . Then, we need to apply the distributive property which states that the multiplication of the form $a\times \left( b+c \right)$ can be written as $a\times b+a\times c$ . Upon comparing this form with our expression, we can say that here,

$\begin{align}

& a=\left( y-7 \right) \\

& b=y \\

& c=-7 \\

\end{align}$

Thus, applying distributive property, the above expression thus becomes,

$\Rightarrow y\times \left( y-7 \right)-7\times \left( y-7 \right)$

We again have terms within the brackets and the brackets are multiplied with another terms. So, we again need to apply the distributive property. For, the first term,

$\begin{align}

& a=y \\

& b=y \\

& c=-7 \\

\end{align}$

So, applying distributive property to the first term, the expression thus becomes,

$\Rightarrow {{y}^{2}}-7y-7\times \left( y-7 \right)$

For the last term, we have

$\begin{align}

& a=7 \\

& b=y \\

& c=-7 \\

\end{align}$

So, applying distributive property to the first term, the expression thus becomes,

$\Rightarrow {{y}^{2}}-7y-7y+49$

Upon simplifying the above expression by adding the two $-7y$ terms, we get,

$\Rightarrow {{y}^{2}}-\left( 2\times 7y \right)+49$

This upon further simplification gives,

$\Rightarrow {{y}^{2}}-14y+49$

**Therefore, we can conclude that the product of the given expression ${{\left( y-7 \right)}^{2}}$ is ${{y}^{2}}-14y+49$**

**Note:**We must be careful while applying the repetitive distributive property as there are a lot of brackets and terms involved and we are most prone to make mistakes here. There is also a predefined formula for the square of subtraction of two terms which is

${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$

Here, in our problem, $a=y,b=7$ . So, we can directly apply this formula.

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