Question

show that 5 + 3 root 5 is an irrational number​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Question -

Show that 5 + 3√5 is an irrational number.

Answer ==>

At first we need to prove that √5 is an irrational number.

Let √5 be Rational number.

Then,

√5 = p/q (where p and q are integers and q is not equal to zero)

Squaring Both sides

(√5)² = (p/q)²

5 = p²/q²

5q² = p²

Here,

we see that

5 divide p

and

5 also divide p².......(¡)

Let p be 5

then,

putting p = 5 on 5q² = p²

we get

5q² = (5)²

5q² = 25

q² = 5

Here,

we see that

5 divide q

And

5 also divide q² ......(¡¡)

from (¡) and (¡¡) we get

5 is a common factor of p and q

so, our assumption is wrong.

√5 is an irrational number.

Proved

Now,

5 + 3√5 = p/q

3√5 = p/q - 5

3√5 = p - 5q/q

√5 = p-5q/3q

Here,

we see that

p-5q/3q is rational number and when we divide two rational number then it is rational but here it is irrational number beacuse we Prove above that √5 is an irrational.

so,

our assumption is wrong .

3 + 3√5 is an irrational number.

Proved .