Question

4. Prove that the following are irrational.
1)
$\frac{1}{ \sqrt{3} }$
2)
$3 + 2 \sqrt{5}$
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Answer 1

Step-by-step explanation:

1)

let us assume that 1/√3 is rarional number

i.e,

1/√3=p/q (where p and q are coprime and q is not equal to 0)

reciprocal

√3/1=q/p

√3=q/p

hence,√3 is a irrational number and q/p is rational number.

√3 is not equal to q/p

so,1/√3 is irrational number

2)

let us assume that 3+2√5 is a rational number

then,

3+2√5=p/q [where p and q are co-prime and q is not equal to 0]

2√5=p/q-3

√5=p/2q - 3/2

hence p and q are integers

√5 is a rational number

but we know That √5 is irrational number

and hence it contradicts our assumption and 3+2√5 is irrational number

hope this will help you,