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nikitadnepr [17]
2 years ago
5

What are some easy ways to find the value of (2017^4−2016^4)/(2017^2+2016^2) without calculator

Mathematics
1 answer:
scoray [572]2 years ago
8 0

Answer:

4033

Step-by-step explanation:

An easy way to solve this problem is to notice the numerator, 2017^4-2016^4 resembles the special product a^2 - b^2. In this case, 2017^4 is a^2 and 2016^4 is b^2. We can set up equations to solve for a and b:

a^2 = 2017^4

a = 2017^2

b^2 = 2016^4

b = 2016^2

Now, the special product a^2 - b^2 factors to (a + b)(a - b), so we can substitute that for the numerator:

<h3>\frac{(2017^2+2016^2)(2017^2 - 2016^2)}{2017^2+2016^2}</h3>

We can notice that both the numerator and denominator contain 2017^2 + 2016^2, so we can divide by \frac{2017^2+2016^2}{2017^2+2016^2} which is just one, and will simplify the fraction to just:

2017^2 - 2016^2

This again is just the special product a^2 - b^2, but in this case a is 2017 and b is 2016. Using this we can factor it:

(2017 + 2016)(2017 - 2016)

And, without using a calculator, this is easy to simplify:

(4033)(1)

4033

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The answer is:      "  3(m − 2)  "  .

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  →  The "factorized version" of the binomial expression, " 3m − 6 " ,  is:

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                →   "  3(m − 2) " .

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Explanation:

To solve:

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<u>Note</u>:  We are given the expression:  " 3m − 6 " ;

and we are asked to "factorize" this expression.

Note that this expression is a "binomial expression" — which means there are two (2) terms — in this case:   "3m" and "6" .

Since only one (1) of these 2 (two) terms has a variable, and the remaining terms is a "constant" (non-zero integer), write these 2 (two) terms as:

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       →   that is:  "3"  ;  and:

 2)  the other term:  "6" .

→  With the numbers:  "3 and 6" , we can factor out a "3" ;  

          →  {since:  " 6 ÷ 3 = 2 "} ;

 →   So;  given:  

               " 3m − 6 " ;  

 →   We can "factor out" a "3" ; as follows:

           →   Take the first term:  " 3m " :

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           →   Take the second term:  "  6 " :

                    "  6 = 3 * (what value?) ;

                 → divide each of this equation by "3" ;

    to isolate the "missing value" on one side of the equation; & to solve for the "missing value" ; as follows:

        →           "  6 / 3 = [ 3 * (what value?) ] / 3  ;

                    →  to get:  " 2  =  "(the missing value)"  ;

    →   So;    "  6 = 3 * (what value?) ;

           →     " 6 = 3 * 2 " .  

           →   Take the second term:  "  6 " :

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__________________________________

So:    " 3m − 6 "  =  (3 * m) − (3 * 2) " .  

Factor out a "3" ;  as follows:

__________________________________

    →  " 3m − 6 "  =  (3 * m) − (3 * 2) " ;

                           =   " 3(m − 2)  "

__________________________________

Let us check our answer:

__________________________________

Note the "distributive property of multiplication"  :

__________________________________

        a(b + c) = ab +  ac ;  

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__________________________________

So:  Take our answer:  " 3(m − 2) " .

__________________________________

" 3(m − 2)  = ?  (3*m)  −  (3*2)  " ??  ;

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 We have:

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     →  " 3(m − 2)  = ?  (3*m)  −  (3*2)  " ??  ;

First let us rewrite this equation;  substitute "4" for "m" ; as follows:

     →   " 3(4 − 2)  = ?  (3*4)  −  (3*2)  " ??  ;

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And calculate;  to see if each side of the equation is equal ; i.e. to see if the equation holds true; as follows:

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     →   "  6  = ?  (6)  " ??  ;

     →   "  6 = 6 " !  Yes!  The equation holds true!

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Hope this answer was helpful!

Best wishes!

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