Q. 185.0( 2 Votes )

# Prove that

(a) is not arational number and

(b) is not arational number.

Answer :

(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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