Prove That √5 + √3 Is an Irrational Number
Video Explanation
Watch the video below for the complete explanation:
Solution
Question: Prove that √5 + √3 is an irrational number.
Proof:
Let us assume that √5 + √3 is a rational number.
Then, squaring both sides, we get:
(√5 + √3)2 is rational
⇒ 5 + 3 + 2√15 is rational
⇒ 8 + 2√15 is rational
Since 8 is a rational number, subtracting 8 from both sides, we get:
2√15 is rational
Dividing both sides by 2 (a non-zero rational number), we get:
√15 is rational
But √15 is irrational because 15 is not a perfect square.
This is a contradiction.
∴ Our assumption is wrong.
Hence, √5 + √3 is an irrational number.
Final Answer
∴ √5 + √3 is an irrational number.
Conclusion
Thus, by the method of contradiction, we have proved that √5 + √3 is irrational.