Question

Show that5+2√7 is an irrational number where√7 is a irrational number

Answer 1

$\huge\mathfrak\green{Answer:}$

Given:

• We have been given a number 5 + 2√7.
• It is also given that √7 is irrational.

To Show:

• We need to show that 5 + 2√7 is an irrational number.

Solution:

Let us assume that 5 + 2√7 is an rational number.

Therefore, 5 + 2√7 can be written in the form of p/q where p and q are integers and q is not equal to zero.

=> 5 + 2√7 = p/q

=> 2√7 = p/q - q

=> √7 = p - 5q/2q

Since, p and q are integers, therefore √7 is rational. But this contradicts the fact that √7 is irrational. Therefore, our assumption was wrong.

Hence, 5 + 2√7 is an irrational number.

Answer 2

$\large\underline\mathrm{Given:-}$

• We have been given a number 5 + 2√7
• It is also given that root 7 is irrational.

$\large\underline\mathrm{To \: find}$

we now to that 5 + 2√7 is an irrational number.

$\large\underline\mathrm{Solution}$

• Let we assume that 5 + 2√7 is an rational number.

Therefore,

5 + 2√7 can in the form of p/q where p and q are integer and q is is not equal to zero.

$\implies$ 5 + 2√7 = p/q

$\implies$ 2√7 = p/q - p

$\implies$ √7 = p - 5q/2q

since, p and q are integer therefore √7 is rational. but contradict, the fact that √7 is irrational. therefore, assumption was wrong.