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vladimir1956 [14]
3 years ago
8

Show that if x and y may both be written as the sum of the squares of two rational integers, then their product xy may also be w

ritten as the sum of the squares of two rational integers.
Mathematics
1 answer:
Georgia [21]3 years ago
7 0

Answer:

See proof below

Step-by-step explanation:

One way to solve this problem is to "add a zero" to complete the required squares in the expression of xy.

Let x=m^2+n^2 and y=l^2+k^2 with m,n,l,k\in \mathbb{Z}. Multiplying the two equations with the distributive law and reordering the result with the commutative law, we get xy=(m^2+n^2)(l^2+k^2)=m^2l^2+m^2k^2+n^2l^2+n^2k^2=n^2l^2+m^2k^2+m^2l^2+n^2k^2

Now, note that 0=2lkmn-2lkmn=2nkml-2nlmk by the commutativity of rational integers. Add this convenient zero the the previous equation to obtain xy=n^2l^2-2nlmk+m^2k^2+m^2l^2+2nkml+n^2k^2=(nl-mk)^2+(ml+nk)^2, thus xy is the sum of the squares of nl-mk,ml+nk\in \mathbb{Z}.

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the ratio of strawberries to grapes in the fruit salad is 3:9. the ratios of strawberries to honey dew is 6:14. if there 36 grap
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Represent strawberry with S, grape with G and honeydew with H

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