Question

Select the correct description for each number set. (8 points)
a) (V 25, V9, v64)
b) (-2.4, 0, 3.1, V81
V
c）（-g,gs）
d) (V12, V11, )
Only Rational Numbers
Only Irrational Numbers
Both Rational & Irrational Numbers

Answer 1

Let's analyze each number set to determine whether the numbers are rational or irrational.

a) \( (\sqrt{25}, \sqrt{9}, \sqrt{64}) \)
- The square roots are:
- \( \sqrt{25} = 5 \) (rational)
- \( \sqrt{9} = 3 \) (rational)
- \( \sqrt{64} = 8 \) (rational)
- **Description:** Only Rational Numbers

b) \( (-2.4, 0, 3.1, \sqrt{81}) \)
- The numbers are:
- \(-2.4\) (rational)
- \(0\) (rational)
- \(3.1\) (rational)
- \(\sqrt{81} = 9\) (rational)
- **Description:** Only Rational Numbers

c) \( (-\sqrt{g}, g^{\frac{1}{2}}) \)
- Without specific values for \( g \):
- If \( g \) is a positive rational number, \(-\sqrt{g}\) is irrational, while \( g^{\frac{1}{2}} \) (the square root) would also be irrational. However, if \( g \) is a perfect square (like 1, 4, 9), both could be rational. But in general:
- \(-\sqrt{g}\) is irrational if \( g \) is not a perfect square.
- \( g^{\frac{1}{2}} \) is irrational if \( g \) is not a perfect square.
- **Description:** Both Rational & Irrational Numbers (depending on the value of \( g \))

d) \( (\sqrt{12}, \sqrt{11}) \)
- The numbers are:
- \(\sqrt{12} = 2\sqrt{3}\) (irrational)
- \(\sqrt{11}\) (irrational)
- **Description:** Only Irrational Numbers

So here are the final descriptions for each number set:

a) Only Rational Numbers
b) Only Rational Numbers
c) Both Rational & Irrational Numbers
d) Only Irrational Numbers