Question

What are some easy ways to find the value of

(2017^4−2016^4)/(2017^2+2016^2) without calculator

Answer 1

4033

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:

`((2017^2+2016^2)(2017^2 - 2016^2))/(2017^2+2016^2)`

We can notice that both the numerator and denominator contain 2017^2 + 2016^2, so we can divide by
`(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