Prove That √2 + √3 Is an Irrational Number
Video Explanation
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Solution
Question: Prove that √2 + √3 is an irrational number.
Proof:
Let us assume that √2 + √3 is a rational number.
Then, squaring both sides, we get:
(√2 + √3)2 is rational
⇒ 2 + 3 + 2√6 is rational
⇒ 5 + 2√6 is rational
Since 5 is a rational number, subtracting 5 from both sides, we get:
2√6 is rational
Dividing both sides by 2 (a non-zero rational number), we get:
√6 is rational
But √6 is irrational because 6 is not a perfect square.
This is a contradiction.
∴ Our assumption is wrong.
Hence, √2 + √3 is an irrational number.
Final Answer
∴ √2 + √3 is an irrational number.
Conclusion
Thus, by the method of contradiction, we have proved that √2 + √3 is irrational.