Why does <msqrt> x </msqrt> <mrow class="MJX-TeXAtom-ORD"> / </mrow> y = <msq

Emmy Knox

Emmy Knox

Answered question

2022-06-16

Why does x / y = x / y / y
Sorry for the awkward title, hard to to sum a mathematical problem with words alone.
Having said that, I recently learned that the root of any value, x, and then that over value y, is identical to the root of x / y / y , as in x / y / y
For e.g, 35 / 7 = 5 / 7 and I cannot logically deduce why. So far, I know that:
35 = 7 × 5 35 7 = 7 × 5 7
And somehow this being equal to:
√(5/7)
So it seems that we divided the numerator, √35 by √5 to give √7. However, 7/√7 is not √7! Yet, apparently it is?
So, a fraction is of course equal to itself if scaled up or down by the same amount, in that 10/2 is the same as 5/1, or 5. So scaling down √35 by √5 makes sense, as it only leaves √7, but it seems that the same cannot be said for the denominator, 7. 7/√7 = √7!? Yet, the two expressions really are the same, √35/7 and √(5/7).
Thanks for any help in advance.
P.S This works for any values, so it's no coincidence.

Answer & Explanation

Jake Mcpherson

Jake Mcpherson

Beginner2022-06-17Added 23 answers

The typesetting is difficult to understand, perhaps this is what you're looking far :
b x y = b x y 2 = b x y 2 .
So for your example with x=35 and y=7 we get :
35 7 = 35 7 2 = 7 5 7 7 = 5 7 .
Sonia Gay

Sonia Gay

Beginner2022-06-18Added 7 answers

I expect you are aware that
a b b = a
because we can cancel the b in the numerator and denominator.
The piece you seem to be missing is that
a a = a a a = a
where in the last step we canceled a from the numerator and denominator.
Or put another way, you know that if x y = z then x = z y . Take x = a , y = a , z = a . Then
a a = a ,
so dividing by a on both sides we get a = a a

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