Q. 185.0( 2 Votes )
(a) is not arational number and
(b) is not arational number.
(a) Let us assume that where a is rational.
Then, squaring both sides-
Using (a + b) 2 =a 2 + 2ab + b 2
Now, a is rational is rational and is an irrational number.
Since a rational number cannot be equal to an irrational number. Our assumption that is rational is wrong.
(b) Let us assume that , where p and q are integers, having no common factors and q ≠ 0.
Taking cube both sides—
7q 3 = p 3 ------(i)
⇒ p is a multiple of 7
Thus p is multiple of 7.
Let p = 7m, where m is an integer.
Then, p 3 = 343 m 3 ------(ii)
⇒ 7q 3 = 343 m 3 [from (i) and (ii)]
⇒ q 3 = 49 m 3 ⇒ q 3 is a multiple of 7.
⇒ q is a multiple of 7.
Thus, p and q are both multiples of 7, or 7 is a factor of p and q.
This contradicts our assumption that p and q have no common factors.
Hence is not a rational number.
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