Question

Show that 5 + √7 is an irrational number.
​

Answer 1

${ \large{ \sf{ \underbrace{\underline{\bigstar \:Answer:}}}}}$

Hence proved that the given $\sqrt 5+ \sqrt 7$

is an irrational number

${ \large{ \boxed{ \pink{ \underline{ \bf \:To\: prove:}}}}}$

To prove whether

$\sqrt 5+ \sqrt 7$

is irrational or not.

$\tt{Solution:}$

Let us assume that

$\sqrt 5+ \sqrt 7$

be rational, and let p/q are co-prime where q is not equal to zero (0).

$\begin{gathered}\begin{array} { l } { \sqrt { 5 } + \sqrt { 7 } = \frac { p } { q } } \\\\ { \sqrt { 5 } = \frac { p } { q } - \sqrt { 7 } } \\\\ { \sqrt { 5 } = \frac { p - \sqrt { 7 } q } { q } } \end{array}\end{gathered}$

We know that

$\sqrt 5$

is irrational while p/q form is rational.

Hence it contradicts our assumption of$\sqrt 5+ \sqrt 7$

is rational.

~ Hence, it is proved that $\sqrt 5+ \sqrt 7$ is irrational.

Done :)

Thankyou :)