which of the following os not irrational? 3+√7 , 3-√7, (3+√7)(3-√7),3√7
Answers
Answer:
Solving First we get,
→ {(3+√7) (3-√7)}
using (a + b)(a - b) = (a² - b²) we get,
→ {(3)² - (√7)²}
Now using , (√a)² = a, we get,
→ (9 - 7)
→ 2
→ (2/1)
→ in the form of p/q, where q ≠ 0.
Therefore,
{(3+√7) (3-√7)} is a rational Number.
________________
Now, Solving second part :-
→ 3√7
→ 3 * √7
→ 3 can be written as (3/1) , where denominator is not ≠ 0 , so a rational number).
→ √7 is a non perfect natural number . As we know that, any non non perfect natural number is always an irrational number).
Therefore,
→ (Rational) * (Irrational)
→ Irrational Number.
Hence, we can conclude that, {(3+√7) (3-√7)} is not an irrational number.
_________________________
Extra :-
Rational + irrational = irrational .
Rational - irrational = irrational .
Rational * irrational = irrational .
Rational ÷ irrational = irrational .
__________________________
Note :- { if we assume all parts of questions different, than, all are irrational numbers as told in extra. } .
Step-by-step explanation:
We have to prove that 3+
7
is irrational.
Let us assume the opposite, that 3+
7
is rational.
Hence 3+
7
can be written in the form
b
a
where a and b are co-prime and b
=0
Hence 3+
7
=
b
a
⇒
7
=
b
a
−3
⇒
7
=
b
a−3b
where
7
is irrational and
b
a−3b
is rational.
Since,rational
= irrational.
This is a contradiction.
∴ Our assumption is incorrect.
Hence 3+ √7
is irrational.
Hence proved.