Question

Check whether the following are rational /irrational give reason why
$7 \sqrt{5}$
$\sqrt{2 + 21}$
$\pi - 131$
$\sqrt{7 + \sqrt{3} }$
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Answer 1

Answer:

7√5 can be written in the form of a/b where a, b are co-prime and b not equal to 0. here √5 is irrational and a/7b is rational number. Therefore, 7√5 is an irrational number. Hence, proved.

√2+21 = √23

23 is not a perfect square values so that, it is an irrational number. ... The decimal expansion of above number is non-terminating non-repeating, So that, it is an irrational number.

π-131 is Rational Number as the Subtraction can be represented in the form of a/b

Lets, assume that √7+√3 is an irrational number,  Then lets equate it by 'x' Therefore,  => x= √7+√3  Squaring both the sides .  => x²=(√7+√3)²  => x²= 10+2√21  Solving them by keeping R.H.S and L.H.S  => x²-10/2= √21 Now,  since,  x²= Rational -10/2= rational...  but, we can say that √21 is an irrational number as it cant be represented it at p/q form,  So, by this method of contradiction,  We contradict that √7+√3 is an irrational number.

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