Prove That ³/_2√5 Is Irrational

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question: Prove that the number ³/_2√5 is irrational.

Step 1: Assume the Contrary

Assume that ³/_2√5 is rational. Then it can be written in the form:

³/_2√5 = p/q,

where p and q are integers having no common factors and q ≠ 0.

Step 2: Rearrange the Equation

Multiply both sides by 2√5:

3 = (2√5 × p) / q

Rearranging, we get:

√5 = ^3q/_2p

Step 3: Square Both Sides

5 = (^3q/_2p)²

5 = ^9q²/_4p²

Multiply both sides by 4p²:

20p² = 9q²

Step 4: Analyze Divisibility

From 20p² = 9q², we see that 9 divides the right-hand side, so 9 must divide 20p².

Since 9 and 20 are coprime, 9 must divide p². Hence, p is divisible by 3.

Let p = 3k, where k is an integer.

Substitute p = 3k:

20(3k)² = 9q²

180k² = 9q²

20k² = q²

This shows that q² is divisible by 5, so q is divisible by 5.

Step 5: Contradiction

Thus, p is divisible by 3 and q is divisible by 5, which contradicts the assumption that p and q have no common factor.

Final Answer

∴ The number ³/_2√5 cannot be expressed as a ratio of two integers. Hence, ³/_2√5 is irrational.

Conclusion

By assuming ³/_2√5 to be rational and reaching a contradiction, we conclude that ³/_2√5 is irrational.