Question

show that 7 - √5 is irrational...given that √5 is irrational​

Answer 1

please mark it as brainliest answer

Answer 2

$\huge{ \bold{ \underline{ \green{ \sf{Detailed \: Answer}}}}}$

• To Prove : $7 - \sqrt{5} \:$ is irrational.

• Given : $\sqrt{5}$ is already irrational.

Solution :

Let us assume that 7 - $\sqrt{5}$ is rational.

$7 - \sqrt{5} = \frac{a}{b}$ ; Where both a & b are rational numbers.

$\sqrt{5} = \frac{a}{b} + 7$

Taking the LCM,

$\sqrt{5} = \frac{a + 7b}{b}$

It is already given that $\sqrt{5}$ is irrational.

And we know that $\frac{ a+ 7b}{b}$ is rational.

Hence, our assumption was wrong.

That means $7 - \sqrt{5}$ is irrational.

$\sf{Hence \: Proved!}$