Question

prove that root 5 is irrational​

Answer 1

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let ,

assume on the contradict that ✓5 is rational number ( where p and q are integers , q≠0 , p and q don't have any common factor except one.

let,

✓5 = p/q

✓5q = p

--> squaring on both sides.

(✓5q)^2 = (p)^2

5q^2 = p^2

here 5 divides p^2

so, 5 divides p also ( according to fundamental theorem of arithmetic )

so,

Now let ,

p = 5k

so,

then,

✓5q = 5k

squaring on both sides .

(✓5q)^2 = (5k)^2

5q^2 = 25k^2

q^2 = 25k^2/5

q^2 = 5k^2

so, 5k^2 = q^2

here,

5 divides q^2

so, 5 divides q also.

Therefore, P and q has a common factor 5

But , according to rational numbers properties p and q should not have any common factors expect 1.

This contradicton is due to that our assumption that ✓5 is rational number is wrong.

Therefore, this contradicts the fact that ✓5 is irrational number.

Hence proved...

Hope this is useful..

Answer 2

Answer:

hi mate.

your answer is,

Step-by-step explanation:

To prove that root 5 is irrational :

Let us assume the opposite,

eg : 5 is rational.

Hence, 5 can be written in the form of a / b.

where a and b ( b is not equal to 0 ) are co - prime ( no common factor other than 1 ) .

hence, root 5 = a / b

= root b = a.

squaring both sides :

( root 5 b square ) = ( a square)

( root 5 b2 ) = a2 .

= 5b2 = a2 .

hence, it is proved that it is irrational.

hope it helps you mate.

please thank and mark my answer as brainliest.