Given:

Question

q equals a positive terminating decimal

m equals a square root of a non-perfect square whole number

Question
Choose whether each statement is "Always," "Sometimes", or "Never" true.

Response area with 6 radio buttons within 2 groups.
Statement True?
3 times q is rational

Always

Sometimes

Never

StartFraction 1 over StartRoot 5 EndRoot EndFraction times m is rational

Always

Sometimes

Never

Answer 1

Let's analyze the statements one by one.

1. Statement: "3 times q is rational"

   • Since \( q \) is a positive terminating decimal, it can be expressed as a fraction (for example, \( q = \frac{a}{10^n} \) for some integers \( a \) and \( n \)). Therefore, multiplying \( q \) by 3, which is also a rational number, will still yield a rational number.
   • Response: Always

2. Statement: "StartFraction 1 over StartRoot 5 EndRoot EndFraction times m is rational"

   • The expression \( \frac{1}{\sqrt{5}} \) is irrational, as the square root of any non-perfect square is irrational.
   • The square root \( m \), as given, is the square root of a non-perfect square whole number, which is also irrational. The product of a rational number (in this case, \( \frac{1}{\sqrt{5}} \)) and an irrational number (like \( m \)) is always irrational.
   • Response: Never

Final Responses:

1. 3 times q is rational: Always
2. StartFraction 1 over StartRoot 5 EndRoot EndFraction times m is rational: Never