Show That 7√5 Is Irrational

Video Explanation

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

Solution

Question: Show that the number 7√5 is irrational.

Step 1: Assume the Contrary

Assume that 7√5 is rational. Then it can be written as a fraction of two integers in lowest terms:

7√5 = p/q, where p and q are integers with no common factors and q ≠ 0.

Step 2: Isolate √5

Rearrange the equation to isolate √5:

√5 = p / (7q)

Now square both sides:

5 = p² / (49q²)

Multiply both sides by 49q²:

5 × 49q² = p²

245q² = p²

Step 3: Analyze Parity and Divisibility

This shows that p² is divisible by 5 (because the left side has factor 5), so p must be divisible by 5.

Let p = 5k for some integer k:

(5k)² = 245q²

25k² = 245q²

k² = 49q²

This shows that k² is divisible by 7², so k is divisible by 7.

Step 4: Contradiction

We found both p has factor 5 and k has factor 7, so p has factors 5 and 7. That means p and q share common factor (since p contains 7 and q remains). This contradicts our assumption that p/q was in lowest terms.

Final Answer

∴ The number 7√5 cannot be written as a ratio of two integers in lowest terms, so 7√5 is irrational.

Conclusion

By assuming 7√5 is rational and reaching a contradiction, we conclude that 7√5 is irrational.