To classify the given numbers into rational and irrational numbers, let's first define them:
- Rational numbers can be expressed as the quotient of two integers (a fraction).
- Irrational numbers cannot be expressed as a fraction; they have non-repeating, non-terminating decimal expansions.
Now, let's analyze each number:
-
\(3\sqrt{27}\):
- \(3\sqrt{27} = 3 \times 3\sqrt{3} = 9\sqrt{3}\) (irrational)
-
\(3\sqrt{28}\):
- \(3\sqrt{28} = 3 \times 2\sqrt{7} = 6\sqrt{7}\) (irrational)
-
\(\pi\):
- \(\pi\) is known to be an irrational number.
-
\(\frac{1}{7}\):
- A ratio of two integers, so it is rational.
-
\(\sqrt{3}\):
- \(\sqrt{3}\) is an irrational number.
-
\(\frac{3}{4}\):
- A ratio of two integers, so it is rational.
-
\(0\):
- \(0\) can be expressed as \(\frac{0}{1}\) (rational).
-
\(\sqrt{25}\):
- \(\sqrt{25} = 5\) (rational).
-
\(\frac{\pi}{2}\):
- Since \(\pi\) is irrational, any non-zero multiple of \(\pi\) is also irrational, so \(\frac{\pi}{2}\) is irrational.
-
\(\sqrt{2}\):
- \(\sqrt{2}\) is an irrational number.
Now, let’s group them:
Rational Numbers:
- \(\frac{1}{7}\)
- \(\frac{3}{4}\)
- \(0\)
- \(\sqrt{25}\) (which is \(5\))
Irrational Numbers:
- \(3\sqrt{27} = 9\sqrt{3}\)
- \(3\sqrt{28} = 6\sqrt{7}\)
- \(\pi\)
- \(\sqrt{3}\)
- \(\frac{\pi}{2}\)
- \(\sqrt{2}\)
So, the final categorization is:
Rational: \(\frac{1}{7}, \frac{3}{4}, 0, \sqrt{25}\)
Irrational: \(3\sqrt{27}, 3\sqrt{28}, \pi, \sqrt{3}, \frac{\pi}{2}, \sqrt{2}\)