Answer :
To find out the irrational number among the given options, let's analyze each option step by step.
Option A: [tex]\(\sqrt{16} - \sqrt{9}\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex]
- [tex]\(\sqrt{9} = 3\)[/tex]
- Therefore, [tex]\(\sqrt{16} - \sqrt{9} = 4 - 3 = 1\)[/tex]
Since 1 is a rational number, Option A is not an irrational number.
Option B: [tex]\(\frac{3}{4}\)[/tex]
- [tex]\(\frac{3}{4}\)[/tex] is already a fraction of two integers (3 and 4), which means it's a rational number.
So, Option B is not an irrational number.
Option C: 0.3333
- 0.3333 is a decimal representation that can be approximated by a fraction, specifically [tex]\(\frac{3333}{10000}\)[/tex].
Since 0.3333 can be represented as a fraction, it is a rational number.
Option D: [tex]\(2 + \sqrt{3}\)[/tex]
- [tex]\(2\)[/tex] is a rational number.
- [tex]\(\sqrt{3}\)[/tex] is an irrational number because it cannot be expressed as a fraction of two integers.
- The sum of a rational number and an irrational number is always irrational.
Thus, [tex]\(2 + \sqrt{3}\)[/tex] is an irrational number.
Given this analysis, the irrational number among the options is:
D) [tex]\(2 + \sqrt{3}\)[/tex]
Hence, the correct answer is [tex]\(4\)[/tex].
Option A: [tex]\(\sqrt{16} - \sqrt{9}\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex]
- [tex]\(\sqrt{9} = 3\)[/tex]
- Therefore, [tex]\(\sqrt{16} - \sqrt{9} = 4 - 3 = 1\)[/tex]
Since 1 is a rational number, Option A is not an irrational number.
Option B: [tex]\(\frac{3}{4}\)[/tex]
- [tex]\(\frac{3}{4}\)[/tex] is already a fraction of two integers (3 and 4), which means it's a rational number.
So, Option B is not an irrational number.
Option C: 0.3333
- 0.3333 is a decimal representation that can be approximated by a fraction, specifically [tex]\(\frac{3333}{10000}\)[/tex].
Since 0.3333 can be represented as a fraction, it is a rational number.
Option D: [tex]\(2 + \sqrt{3}\)[/tex]
- [tex]\(2\)[/tex] is a rational number.
- [tex]\(\sqrt{3}\)[/tex] is an irrational number because it cannot be expressed as a fraction of two integers.
- The sum of a rational number and an irrational number is always irrational.
Thus, [tex]\(2 + \sqrt{3}\)[/tex] is an irrational number.
Given this analysis, the irrational number among the options is:
D) [tex]\(2 + \sqrt{3}\)[/tex]
Hence, the correct answer is [tex]\(4\)[/tex].
