Doubt by Aman
Solution :
Let us consider that √13-√17 is a rational number. So it can be written in the form of a/b where a and b are co-prime positive integers.
√13-√17 = a/b
√13 =
Squaring both sides.
Let us consider that √13-√17 is a rational number. So it can be written in the form of a/b where a and b are co-prime positive integers.
√13-√17 = a/b
√13 =
Squaring both sides.
(√13)² = (a/b + √17)²
13 = (a/b)²+(√17)²+2(a/b)(√17)
13 = a²/b² + 17+2a√17/b
13-17 = a²/b²+2a√17/b
-4 = a²/b²+2ab√17/b²
-4 = (a²+2ab√17)/b²
-4b²=a²+2ab√17
-4b²-a²-2ab=√17
-(a²+4b²+2ab)=√17
∵ a and b are integers.
∴ -(a²+4b²+2ab) is rational.
13 = (a/b)²+(√17)²+2(a/b)(√17)
13 = a²/b² + 17+2a√17/b
13-17 = a²/b²+2a√17/b
-4 = a²/b²+2ab√17/b²
-4 = (a²+2ab√17)/b²
-4b²=a²+2ab√17
-4b²-a²-2ab=√17
-(a²+4b²+2ab)=√17
∵ a and b are integers.
∴ -(a²+4b²+2ab) is rational.
⇒ √17 is also a rational number. But we know that √17 is irrational.
It means, our assumption was wrong.
Hence, √13-√17 is irrational.
It means, our assumption was wrong.
Hence, √13-√17 is irrational.
Short Answer :
If this is asked in 1 mark question then you can write this :
We know, √13 and √17 and are irrational and the subtraction of two irrational number is always irrational.
Hence √13-√17 is irrational.
If this is asked in 1 mark question then you can write this :
We know, √13 and √17 and are irrational and the subtraction of two irrational number is always irrational.
Hence √13-√17 is irrational.