Could someone help with this math question?

Question

Ok. I'm seriously confused and i need some help. any answers or instructions would be greatly thanked!

Find three numbers w,x and y, none of which are perfect squares or zero, that make the following relationship true

√w + √x = √y

Answer 1

i think you are looking for w,x,y to be integer number or else question is pointless

expand the relationship

( √w + √ x )^2 = (√y) ^2

w + 2√wx +x = y

in order for y to be an integer number w + 2√wx +x must add up to be integer

w and x are integers itself so they are fine

the only problem is 2√wx

to make this a integer wx must be a perfect square

now we can see that their are infinite number of possible solutions

the easiest way to make wx a perfect square is to multiply 2 exact same numbers together

so lets assume w and x are equal

√w + √x = √y

√w + √w  = √y

2√w = √y

square both sides

4w = y

as long as w is not zero or a perfect square

so one solution of that is 2

w = 2 x= 2 and y = 8 another solution is w= 3 x = 3  y = 12

now lets find solution where w and x are different integers

remember the part where 2√wx must also be integer

this means wx is perfect square

so one way of getting different numbers is to use a prime (we use prime to ensure the number w.x.y are not perfect squares) number multiplied by its cube (we use cube because no cubes are perfects squares except 1^3 which is not allowed anyways since it isn't prime  also as sidenote a number of multiply by its cube equals the number with exponent of four which allows for wx to be perfect square! ) this works since the exponents will add up when multiplied  (exponent law) to get even number

one example is w= 2 (as prime) and x=2^3 = 8 (as its cube)

and y will equal 18

there are many other ways to do it but these are the simpler ones

now you can see there are many different ways of satisfying relation, in fact this type of equation is known as a diphonatine equation where there are infinite solution sets and only integers are allowed

hope this helps

Answer 2

ok, I don't quite know how this worked but...  ÃƒÂ¢Ã‚ˆÂš2 + √2 = 2.8284... which when squared gives 8 so √2 + √2 = √8.  That is if you can have two of the 3 numbers being equal.  Not sure how that works...

Wow...  if you add the squareroots of two nonperfect squares (call this number a), then the answer is √(a*4)  so like √3 + √3 = √12, √5 + √5 = √20 and so on.  How interesting...  Learn something new every day.  Not sure if this fully answered your question but it never said 2 of the numbers couldn't be equal, just that they couldn't be perfect squares or zero.

Hope this helped a little

Answer 3

√w + √x = √y

Square both sides:

w + x + 2√(wx) = y

wx must be a perfect square

Try 2 and 8 (2×8 = 16 = 4²)

w = 2

x = 8

2√(wx) = 2×4 = 8

y = 2 + 8 + 8  = 18

None of 2, 8 or 18 are perfect squares...

√2 + √8 = √18

This is just one possibility.