To determine which numbers are irrational, let's analyze each one step-by-step:
### Number 1: [tex]\(-2.3456\)[/tex]
- This is a finite decimal.
- Finite decimals are rational because they can be expressed as a fraction (e.g., [tex]\(-2.3456 = - \frac{23456}{10000}\)[/tex]).
- Therefore, [tex]\(-2.3456\)[/tex] is a rational number.
### Number 2: [tex]\(\frac{\pi}{4}\)[/tex]
- [tex]\(\pi\)[/tex] (pi) is an irrational number.
- Dividing an irrational number by a rational number (here, [tex]\(\frac{1}{4}\)[/tex], or 4) still results in an irrational number.
- Thus, [tex]\(\frac{\pi}{4}\)[/tex] is an irrational number.
### Number 3: [tex]\(\sqrt[3]{9}\)[/tex]
- The cube root of 9 cannot be expressed as a fraction.
- Since [tex]\(9\)[/tex] is not a cube of any rational number, [tex]\(\sqrt[3]{9}\)[/tex] remains irrational.
- Therefore, [tex]\(\sqrt[3]{9}\)[/tex] is an irrational number.
### Number 4: [tex]\(2 + \sqrt{16}\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex], which is a rational number.
- Adding two rational numbers [tex]\(2 + 4 = 6\)[/tex], we get another rational number.
- Thus, [tex]\(2 + \sqrt{16}\)[/tex] is a rational number.
Based on this analysis, the irrational numbers from the options provided are:
### Answer Options:
1. [tex]\(-2.3456\)[/tex] and [tex]\(2+\sqrt{16}\)[/tex] - Both are rational.
2. [tex]\(-2.3456\)[/tex] and [tex]\(\frac{\pi}{4}\)[/tex] - First is rational, second is irrational.
3. [tex]\(\sqrt[3]{9}\)[/tex] and [tex]\(2+\sqrt{16}\)[/tex] - First is irrational, second is rational.
4. [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\(\sqrt[3]{9}\)[/tex] - Both are irrational.
The correct answer is:
- [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\(\sqrt[3]{9}\)[/tex]
Therefore, the result is:
### [tex]\(\boxed{-1}\)[/tex]
