Question

Show that under root 3 + under root 5 is irrational

Answer 1

We have to show that √3+√5 is irrational
We would prove it ny the contradiction method:-

If possible let √3+√5 be a rational number equal to a :-

√3+√5=a
(√3+√5)^2=a^2. (squaring both the sides)
3+2√15+5=a^2
8+2√15=a^2
√15=(a^2-8)/2

So, here a is rational se a^2 will be rational so
(a^2-8)/2 will be rational
So, √15 will be rational because √15=(a^2-8)/2

But it contradicts the fact that √15 is irrational

So our assumption is wrong that √3+√5 is rational

So √3+√5 is irrational

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Answer 2

Heya.......!!!!

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To prove :- √3 + √5 = irrational .

Let there be a number ' y ' that is equal to √3 + √5

here y is rational number .

=> y = √3 + √5

squaring both the sides.

=> y​^2 = 3 +5 + 2√15
=> y^2 = 8 + 2√15
=> y^2 -8 = 2√15

=> ( y^2 - 8 ) / 2 = √15


Now see the RHS and LHS carefully.

In RHS the number ' y ' is rational then y^2 is also be rational and numbers are also rational . So we can say that RHS is rational. But on LHS √15 is irrational and RHS = LHS

Hence it's contradiction.

Thus , √3 + √5 is irrational .



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