Question

prove that 7+3under root 17 is irrational​

Answer 1

Step-by-step explanation:

First, we would have to prove that √17 is irrational.

Prove by contradiction :

Suppose, √17 is a rational number.

Which implies that it could be obtained by the division of Natural Numbers.

Let p/q = √17 for some integers p and q and q not equal to 0

WLOG, p, q > 0 and p, q DO NOT have any common factors other than 1

On squaring both sides, we get :

p^2 = 17*q^2 _________ Equation 1

Since p^2 is a multiple of 17 and 17 is a prime, p is also a multiple of 17

Let k = p/17, p = 17k

Substituting the value of p in Equation 1, we get :

(17k)^2 = 17*q^2 or,

17*k^2 = q^2 _________ Equation 2

Since q^2 is a multiple of 17 and 17 is a prime, q is also a multiple of 17

But this contradicts our assumption that p and q DO NOT have any common factors other than 1

Therefore, √17 is irrational.

Now, An irrational number divided by a rational number, would result in an Irrational number.

So, √17/3 is irrational.

Subsequently, 7 + √17/3 is irrational, since addition with irrational numbers, gives irrational number.