Question

prove that 5-3√15 is a irrational number.. ​

Answer 1

Answer:

Let 5-3√15 is a rational number i.e,

$5 - 3 \sqrt{15} = \frac{p}{q} (q \: is \: not \: equal \: to \: 0) \\ \\ 3 \sqrt{15} = 5 - \frac{p}{q} \\ \\ 3 \sqrt{15} = \frac{5q - p}{q} \\ \\ \sqrt{15} = \frac{5q - p}{3q}$

LHS is irrational number whereas RHS is rational.

Which is contradiction.

Therefore our supposition is wrong.

So 5-3√15 is irrational number.

Answer 2

${\blue{\underline{\underline{\bold{\huge{Solution:-}}}}}}$

Let us assume that 5-3√15 is rational.

That's why, we can write it in the form of p/q.

Here, p & q are co- prime and q ≠ 0.

⟹ 5 -3√15 = p/q

⟹ -3√15 = p/q - 5

⟹ √15 = (p-5q)/-3q

⟹ √15 = (5q-p)/3q

As p & q both are integer.....

Therefore, (5q-p)/3q is a rational number.

But √15 is a irrational number.

✰We know that, a rational number can never be equal to irrational number.

☞ Hence, √15 ≠ (5q-p)/3q

✰So, our contradiction is wrong &

5 -3√15 is not a rational number.

☞ It’s a irrational number.

Hence, ( proved)

${\huge{\mathcal{\pink{Hope \ It's \ Helpful..!!!}}}}$