Question

prove that √3-√5 is an irrational number with clear explanation​

Answer 1

Answer:

the above is the solution

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Answer 2

Answer:

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Step-by-step explanation:

Let us assume it to be a rational number.

Rational numbers are the ones that can be expressed in p/q form where p,q are integers and q isn't equal to zero.

√3-√5=p/q

√3=p/q-√5

squaring on both sides,

3=(p/q - 5)^2

3=p^2/q^2 +5-2×√5×p/q

(2√5p)/q=(p/q)^2-3+5

2√5(p/q)=2+(p/q)^2

√5={2+(p/q)^2}/{√5(p/q)}

√5={(2q^2+p^2)(q)}/{(q^2)(2p)}

√5=(2q^2+p^2)/2pq

As p and q are integers RHS is also rational.

As RHS is rational LHS is also rational (i.e ) √5 is rational.

But this contradicts the fact that √5 is irrational.

This contradiction arose because of our false assumption.

so, √3-√5 is irrational.