Question

prove that 2 + 3 root 3 is an irrational number when it is given that root 3 is an irrational number​

Answer 1

$let \: 2 + 3 \sqrt{3} \: be \: a \: rational \: no \\ then \: \\ 2 + 3 \sqrt[]{3} = \frac{p}{q} \\ so \: \: \sqrt{3} = \frac{p - 2q}{3q } \\ as \: p - 2q \div 3q \: is \: rational$

so ,/3 must also be rational

but it contradict the fact that it is irrational

so our assumption is wrong

$2 + \sqrt[3]{3} \: is \: irrational$

Answer 2

$2 + 3 \sqrt3$ is an irrational number.

Step-by-step explanation:

Rational number: Which can be defined as p/q, where p and q are integers and q≠0.

Irrational number : Which can not be defined as p/q, where p and q are integers and q≠0.

It is given that root 3 is an irrational number​

Let $2 + 3 \sqrt3$ is a rational number.

$2 + 3 \sqrt3=\dfrac{p}{q}$

where, p and q are integers and q≠0.

$3 \sqrt3=\dfrac{p}{q}-2$

$3 \sqrt3=\dfrac{p-2q}{q}$

$\sqrt3=\dfrac{p-2q}{3q}$

It means √3 is a rational number. Which contradict the given statement.

It means $2 + 3 \sqrt3$ is an irrational number.

Hence proved.

#Learn more

Prove root 3 is an irrational number.

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