Prove That (5 + 3√2) Is an Irrational Number

Video Explanation

Watch the video below for the complete explanation:

Solution

Question: Given that √2 is irrational, prove that (5 + 3√2) is an irrational number.

Proof:

Let us assume that (5 + 3√2) is a rational number.

Since 5 is a rational number, subtracting 5 from both sides, we get:

3√2 = (5 + 3√2) − 5

This implies that 3√2 is a rational number.

Dividing both sides by 3 (which is a non-zero rational number), we get:

√2 is a rational number.

But this contradicts the given fact that √2 is irrational.

∴ Our assumption is wrong.

Hence, (5 + 3√2) is an irrational number.

Final Answer

∴ (5 + 3√2) is an irrational number.

Conclusion

Thus, using the method of contradiction and the fact that √2 is irrational, we have proved that (5 + 3√2) is irrational.