Answer :
To determine which numbers are irrational, we need to analyze each of the given numbers:
1. Number: [tex]\( -2.3456 \)[/tex]
- This number has a finite decimal representation.
- Numbers with finite decimal representations are rational.
- Therefore, [tex]\( -2.3456 \)[/tex] is not irrational.
2. Number: [tex]\( \frac{\pi}{4} \)[/tex]
- The number [tex]\(\pi\)[/tex] is well-known to be irrational.
- When you divide an irrational number by a rational, the result remains irrational.
- Therefore, [tex]\( \frac{\pi}{4} \)[/tex] is irrational.
3. Number: [tex]\( \sqrt[3]{9} \)[/tex] (the cube root of 9)
- To determine if [tex]\( \sqrt[3]{9} \)[/tex] is rational or irrational, consider this:
- [tex]\( \sqrt[3]{9} \)[/tex] is not an integer or a fraction that can be expressed in the form [tex]\( \frac{a}{b} \)[/tex] where a and b are integers and b is not zero.
- Therefore, [tex]\( \sqrt[3]{9} \)[/tex] is an irrational number.
4. Number: [tex]\( 2 + \sqrt{16} \)[/tex]
- Calculate [tex]\( \sqrt{16} \)[/tex], which equals 4.
- Thus, [tex]\( 2 + \sqrt{16} = 2 + 4 = 6 \)[/tex].
- The number 6 is a rational number because it can be written as [tex]\( \frac{6}{1} \)[/tex].
- Therefore, [tex]\( 2 + \sqrt{16} \)[/tex] is not irrational.
With this analysis in mind, we summarize which numbers are irrational:
- List of Irrational Numbers: [tex]\( \frac{\pi}{4} \)[/tex] and [tex]\( \sqrt[3]{9} \)[/tex].
Now, we match these findings with the given options:
- Option 1: [tex]\(-2.3456\)[/tex] and [tex]\(2 + \sqrt{16}\)[/tex]
- Both numbers are rational, so this is not the correct choice.
- Option 2: [tex]\(-2.3456\)[/tex] and [tex]\(\frac{\pi}{4}\)[/tex]
- Only one of the numbers is irrational, so this is not the correct choice.
- Option 3: [tex]\(\sqrt[3]{9}\)[/tex] and [tex]\(2 + \sqrt{16}\)[/tex]
- Only one of the numbers is irrational, so this is not the correct choice.
- Option 4: [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\(\sqrt[3]{9}\)[/tex]
- Both numbers are irrational, so this is the correct choice.
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{\pi}{4} \text{ and } \sqrt[3]{9}} \][/tex]
1. Number: [tex]\( -2.3456 \)[/tex]
- This number has a finite decimal representation.
- Numbers with finite decimal representations are rational.
- Therefore, [tex]\( -2.3456 \)[/tex] is not irrational.
2. Number: [tex]\( \frac{\pi}{4} \)[/tex]
- The number [tex]\(\pi\)[/tex] is well-known to be irrational.
- When you divide an irrational number by a rational, the result remains irrational.
- Therefore, [tex]\( \frac{\pi}{4} \)[/tex] is irrational.
3. Number: [tex]\( \sqrt[3]{9} \)[/tex] (the cube root of 9)
- To determine if [tex]\( \sqrt[3]{9} \)[/tex] is rational or irrational, consider this:
- [tex]\( \sqrt[3]{9} \)[/tex] is not an integer or a fraction that can be expressed in the form [tex]\( \frac{a}{b} \)[/tex] where a and b are integers and b is not zero.
- Therefore, [tex]\( \sqrt[3]{9} \)[/tex] is an irrational number.
4. Number: [tex]\( 2 + \sqrt{16} \)[/tex]
- Calculate [tex]\( \sqrt{16} \)[/tex], which equals 4.
- Thus, [tex]\( 2 + \sqrt{16} = 2 + 4 = 6 \)[/tex].
- The number 6 is a rational number because it can be written as [tex]\( \frac{6}{1} \)[/tex].
- Therefore, [tex]\( 2 + \sqrt{16} \)[/tex] is not irrational.
With this analysis in mind, we summarize which numbers are irrational:
- List of Irrational Numbers: [tex]\( \frac{\pi}{4} \)[/tex] and [tex]\( \sqrt[3]{9} \)[/tex].
Now, we match these findings with the given options:
- Option 1: [tex]\(-2.3456\)[/tex] and [tex]\(2 + \sqrt{16}\)[/tex]
- Both numbers are rational, so this is not the correct choice.
- Option 2: [tex]\(-2.3456\)[/tex] and [tex]\(\frac{\pi}{4}\)[/tex]
- Only one of the numbers is irrational, so this is not the correct choice.
- Option 3: [tex]\(\sqrt[3]{9}\)[/tex] and [tex]\(2 + \sqrt{16}\)[/tex]
- Only one of the numbers is irrational, so this is not the correct choice.
- Option 4: [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\(\sqrt[3]{9}\)[/tex]
- Both numbers are irrational, so this is the correct choice.
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{\pi}{4} \text{ and } \sqrt[3]{9}} \][/tex]
