Prove that 5-3 /7 root 3 is irrational
Answers
Answer:
5−
7
3
3
is an irrational number.
Step-by-step explanation:
To proof : 5-\frac{3}{7\sqrt{3}}5−
7
3
3
is a irrational number ?
Proof :
Let us assume that,
5-\frac{3}{7\sqrt{3}}5−
7
3
3
is a rational number.
Then it can be written in p/q form where p and q are co-prime.
\frac{3}{7\sqrt{3}}=5-\frac{p}{q}
7
3
3
=5−
q
p
\frac{3}{7\sqrt{3}}=\frac{5q-p}{q}
7
3
3
=
q
5q−p
\frac{3q}{5q-p}=7\sqrt{3}
5q−p
3q
=7
3
\frac{3q}{7(5q-p)}=\sqrt{3}
7(5q−p)
3q
=
3
Here, 7,5,3,q,and p are integers.
i.e. LHS is rational number.
We know, \sqrt{3}
3
is an irrational number.
A rational number cannot be equal to irrational number.
This contradict the assumption.
Therefore, 5-\frac{3}{7\sqrt{3}}5−
7
3
3
is an irrational number.
#Learn more
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