Question

Prove that the product of the two rational numbers 36−−√ and 53 is a rational number. Find the product and explain or show your work to prove the product of two rational numbers is a rational number.

Answer 1

Interpretation note: √36 = 6, so it is rational. The other number could be 53 (the integer fifty‑three) or 5/3 (five‑thirds). Either way the product is rational; I show both.

General proof that the product of two rationals is rational:
- A rational number can be written as a/b with integers a, b and b ≠ 0.
- Let x = a/b and y = c/d be rational with integers a,b,c,d and b,d ≠ 0.
- Then xy = (a/b)(c/d) = (ac)/(bd). Since ac and bd are integers and bd ≠ 0, xy is rational.

Now apply this to the given numbers:
1. If the numbers are √36 and 53:
- √36 = 6 = 6/1 (rational). 53 = 53/1.
- Product = 6 · 53 = 318 (which is 318/1), a rational number.

2. If the numbers are √36 and 5/3:
- √36 = 6.
- Product = 6 · (5/3) = 30/3 = 10, a rational number.

Thus in either case the product is rational (318 or 10), consistent with the general proof.