Question

What must be true about ab
if cd
and ad+bcbd
are both rational numbers? Enter 1 for option 1 and 2 for option 2.

Option 1: ab
is rational

Option 2: ab
is irrational

(1 point)

Answer 1

To determine the relationship between \( ab \) and the given conditions, we analyze the implications of \( cd \) and \( ad + bcbd \) being rational numbers.

Assume:

1. \( cd \) is rational.
2. \( ad + bcbd \) is rational.

From these statements, we need to conclude something about \( ab \).

Let’s analyze the implications:

- If both \( cd \) (which is a product of two terms) and \( ad + bcbd \) (which is a sum involving products) are rational, this typically suggests a relationship between the terms \( a, b, c, d \).

For \( ab \) to be rational, we need to consider how it interacts with \( a \) and \( b \) separately:
- If \( a \) is rational, then \( ab \) being rational would depend on \( b \) being rational.
- If \( a \) is irrational, \( b \) would also need to be irrational (in a compatible way) for \( ab \) to possibly be rational.

However, the only way we can firmly conclude a specific case is to recognize that both conditions involving products and sums can often lead to \( ab \) being rational when intertwined with rational products and sums.

Given that nothing directly contradicts \( ab \) being rational while both provided conditions are satisfied, we conclude:

**Option 1: \( ab \) is rational**.

So, the answer is **1**.