Question

How do you simplify \[(x+7)(x-7)\]?

Answer 1

Hint: Let a monomial be in the form \[a{{x}^{p}}\], where a is a constant coefficient and p is a constant power. In case of multiplying two monomials together: \[\Rightarrow A{{x}^{p}}={{a}_{1}}{{x}^{{{p}_{1}}}}\times {{a}_{2}}{{x}^{{{p}_{2}}}}\]. The coefficients will multiply and powers will sum then, \[A={{a}_{1}}{{a}_{2}}\] and \[p={{p}_{1}}+{{p}_{2}}\]. Hence, \[\Rightarrow A{{x}^{p}}={{a}_{1}}{{a}_{2}}{{x}^{{{p}_{1}}}}{{x}^{{{p}_{2}}}}={{a}_{1}}{{a}_{2}}{{x}^{{{p}_{1}}+{{p}_{2}}}}\]. Similarly, in case of multiplying monomials by binomials also we do the same thing for each term.

Complete step by step answer:
As per the question, we need to simplify \[(x+7)(x-7)\]. We need to distribute the first binomial to the second binomial. By splitting up the equation, we can expand \[(x+7)(x-7)\] using the distributive property given by \[(a+b)(c+d)=a(c+d)+b(c+d)\] into the following equation:
\[\Rightarrow x(x-7)+7(x-7)\]
We know that multiplication of x by x is equal to \[{{x}^{2}}\] and multiplication of x by 7 is equal to \[7x\] and multiplication of 7 by 7 gives 49. Now, by substituting all these terms into the previous equation, we get the equation as the sum of \[{{x}^{2}}\],\[-7x\],\[7x\] and \[-49\].
We can express the obtained equation as
\[\Rightarrow {{x}^{2}}-7x+7x-49\]
By adding \[-7x\] and \[7x\], we get 0. Then the equation will be
 \[\Rightarrow {{x}^{2}}-49\]
\[\therefore {{x}^{2}}-49\] is the required simplified form of the given equation \[(x+7)(x-7)\].

Note:
While solving such types of questions, we need to take care while calculating the product of two monomials. While calculating, we need to concentrate on the calculations of coefficient product and also about the summation of powers to get the resulting monomial. Here also we follow PEMDAS rule while simplifying the equation. In some cases, we need to rearrange the terms according to their powers, the monomial with highest power needs to be written first followed by its next highest power till constant (in descending order).