Question

prove that ✓6+✓2 is irrational number ?​

Answer 1

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prove that ✓6+✓2 is irrational number ?

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Let us assume that √6+√2 is a rational number then we can write in the form of p/q, where p and q are co-primes and q is not equal to 0

$√6+√2=\frac{p}{q}$

$squaring \: on \: both \: sides$

$(√6+√2)² = \frac{p²}{q²}$

$6+2+2√8 =\frac{p^{2} }{q^{2} }$

$8+2√8=\frac{p²}{q²}$

$2√8=\frac{p²}{q²}-8$

$√8=\frac{p²}{q²}-8-2$

$√8=\frac{p²}{q²}-10$

$hence \: \frac{p²-}{q²}-10 \: is \: an \: rational \: number$

$But \: we \: know \: that \: √8 \: is \: irrational \: number$

$From \: this \: our \: assumption \: is \: wrong$

$Therefore, \: √6 + √2 \: is \: not \: an \: rational \: number$

$Hence, \: √6+√2 \: is \: an \: irrational \: number$

$Hence \: proved$

[Hope this helps you.../]