Question

Root3 +root5 find irrational number

Answer 1

6.7320..................

Answer 2

ANSWER:

• (√3+√5) is an Irrational number.

GIVEN:

• Number = √3+√5

TO FIND:

• (√3+√5) is an Irrational number.

SOLUTION:

Let (√3+√5) be a rational number which can be expressed in the form of p/q where p and q have no other common factor than 1.

$\implies \sqrt{3} + \sqrt{5} = \dfrac{p}{q} \\ \\ \implies \: \sqrt{3} = \dfrac{p}{q} - \sqrt{5} \\ \\ \: \: \: squaring \: both \: sides \: we \: get. \\ \\ \implies \: (\sqrt{3} ) {}^{2} = ( \dfrac{p}{q} - \sqrt{5} ) {}^{2} \\ \\ \implies \: 3 = \dfrac{p {}^{2} }{q {}^{2} } + 5 - \frac{2 \sqrt{5}p }{q} \\ \\ \implies \: \dfrac{2 \sqrt{5} p}{q} = \dfrac{p {}^{2} }{q {}^{2} } + 2 \\ \\ \implies \: \frac{2 \sqrt{5} p}{q} = \frac{p {}^{2} + 2q {}^{2} }{q {}^{2} } \\ \\ \implies \: \sqrt{5} = \frac{p {}^{2} + 2q {}^{2} }{2pq}$

Here:

• (p²+2q²)/2pq is rational but √5 is Irrational
• Thus our contradiction is wrong.
• (√3+√5) is an Irrational number.