Prove That 2 + 5√3 Is an Irrational Number

Video Explanation

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Solution

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

Proof:

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

Since 2 is a rational number, subtracting 2 from both sides, we get:

5√3 = (2 + 5√3) − 2

This implies that 5√3 is a rational number.

Dividing both sides by 5 (which is a non-zero rational number), we get:

√3 is a rational number.

But this contradicts the given fact that √3 is an irrational number.

∴ Our assumption is wrong.

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

Final Answer

∴ 2 + 5√3 is an irrational number.

Conclusion

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