Sure! Let's go through each option to determine which one produces an irrational number when added to [tex]\(\frac{1}{4}\)[/tex].
1. Option A: [tex]\(3.4641016 \ldots\)[/tex]
- [tex]\(3.4641016 \ldots\)[/tex] is approximately the square root of 12, which is irrational.
- When you add an irrational number to a rational number, the result is irrational.
- Therefore, [tex]\(\frac{1}{4} + 3.4641016 \ldots\)[/tex] is irrational.
2. Option B: [tex]\(-\frac{4}{9}\)[/tex]
- [tex]\(-\frac{4}{9}\)[/tex] is a rational number.
- When you add two rational numbers, the result is rational.
- Therefore, [tex]\(\frac{1}{4} + -\frac{4}{9}\)[/tex] is rational.
3. Option C: [tex]\(0.545454 \ldots\)[/tex]
- [tex]\(0.545454 \ldots\)[/tex] is a repeating decimal, which means it is a rational number.
- When you add two rational numbers, the result is rational.
- Therefore, [tex]\(\frac{1}{4} + 0.545454 \ldots\)[/tex] is rational.
4. Option D: [tex]\(\sqrt{4}\)[/tex]
- [tex]\(\sqrt{4} = 2\)[/tex], which is a rational number.
- When you add two rational numbers, the result is rational.
- Therefore, [tex]\(\frac{1}{4} + \sqrt{4}\)[/tex] is rational.
From this analysis, the number that produces an irrational number when added to [tex]\(\frac{1}{4}\)[/tex] is:
A. [tex]\(3.4641016 \ldots\)[/tex]
