(i) 3+5
Let us assume that 3+5 is a rational number, say r.
Then,
3+5=r⇒5=r−3
As r is rational, r - 3 is rational
⇒5 is rational
But this contradicts the fact that 5 is irrational.
Hence, our assumption is wrong.
∴ 3+5 is an irrational number.
(ii) 15−27
Let us assume that 15−27 is a rational number, say r.
Then,
15−27=r⇒27=15−r⇒7=215−r
As r is rational, 15 - r is rational
⇒215−r is rational
⇒7 is rational
But this contradicts the fact that 7 is irrational.
Hence, our assumption is wrong.
∴ 15−27 is an irrational number.
(iii) 3−51
Let us rationalise the denominator
3−51=3−51×3+53+5=(3)2−(5)23+5=9−53+5=43+5
Let us assume that 43+5 is a rational number, say r.
Then,
43+5=r⇒3+5=4r⇒5=4r−3
As r is rational, 4r is rational
⇒ 4r - 3 is also rational
⇒5 is rational
But this contradicts the fact that 5 is irrational.
Hence, our assumption is wrong.
⇒43+5 is an irrational number.
∴ 3−51 is an irrational number.