Question

How do you find the product of ${{\left( y-7 \right)}^{2}}$ ?

Answer 1

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,
$\begin{aligned} & a=\left( y-7 \right) \\\ & b=y \\\ & c=-7 \\\ \end{aligned}$
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,
$\begin{aligned} & a=y \\\ & b=y \\\ & c=-7 \\\ \end{aligned}$
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
$\begin{aligned} & a=7 \\\ & b=y \\\ & c=-7 \\\ \end{aligned}$
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.