Question

prove that root 5 + root 7 is an irrational number.

Answer 1

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Answer 2

Answer:

Hence proved that the given $\sqrt 5+ \sqrt 7$ is an irrational number

To prove:

To prove whether $\sqrt 5+ \sqrt 7$ is irrational or not.

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).

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

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.