Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Let's analyze each of the options to determine which number produces an irrational number when multiplied by \(\frac{1}{4}\).
### Option A: \(\sqrt{12}\)
1. The square root of 12 is \(\sqrt{12}\).
2. When we multiply \(\sqrt{12}\) by \(\frac{1}{4}\), we get:
[tex]\[ \frac{\sqrt{12}}{4} \][/tex]
3. Since \(\sqrt{12}\) is an irrational number (it cannot be expressed as a simple fraction), \(\frac{\sqrt{12}}{4}\) is also an irrational number, because multiplying an irrational number by a rational one (non-zero) remains irrational.
### Option B: \(-\sqrt{36}\)
1. The square root of 36 is \(\sqrt{36} = 6\), so \(-\sqrt{36} = -6\).
2. When we multiply \(-6\) by \(\frac{1}{4}\), we get:
[tex]\[ -6 \times \frac{1}{4} = -\frac{6}{4} = -\frac{3}{2} \][/tex]
3. \(-\frac{3}{2}\) is a rational number (it can be expressed as a simple fraction).
### Option C: \(0.444444 \ldots\)
1. Express \(0.444444 \ldots\) as a fraction, which is \(\frac{4}{9}\) (since it is a repeating decimal).
2. When we multiply \(\frac{4}{9}\) by \(\frac{1}{4}\), we get:
[tex]\[ \frac{4}{9} \times \frac{1}{4} = \frac{4 \times 1}{9 \times 4} = \frac{4}{36} = \frac{1}{9} \][/tex]
3. \(\frac{1}{9}\) is a rational number (it can be expressed as a simple fraction).
### Option D: \(\frac{4}{3}\)
1. \(\frac{4}{3}\) is already in fractional form.
2. When we multiply \(\frac{4}{3}\) by \(\frac{1}{4}\), we get:
[tex]\[ \frac{4}{3} \times \frac{1}{4} = \frac{4 \times 1}{3 \times 4} = \frac{4}{12} = \frac{1}{3} \][/tex]
3. \(\frac{1}{3}\) is a rational number (it can be expressed as a simple fraction).
### Conclusion
Among the given options, the number that produces an irrational number when multiplied by \(\frac{1}{4}\) is \(\sqrt{12}\).
Thus, the correct answer is:
[tex]\[ \boxed{1 \text{ (A)}} \][/tex]
### Option A: \(\sqrt{12}\)
1. The square root of 12 is \(\sqrt{12}\).
2. When we multiply \(\sqrt{12}\) by \(\frac{1}{4}\), we get:
[tex]\[ \frac{\sqrt{12}}{4} \][/tex]
3. Since \(\sqrt{12}\) is an irrational number (it cannot be expressed as a simple fraction), \(\frac{\sqrt{12}}{4}\) is also an irrational number, because multiplying an irrational number by a rational one (non-zero) remains irrational.
### Option B: \(-\sqrt{36}\)
1. The square root of 36 is \(\sqrt{36} = 6\), so \(-\sqrt{36} = -6\).
2. When we multiply \(-6\) by \(\frac{1}{4}\), we get:
[tex]\[ -6 \times \frac{1}{4} = -\frac{6}{4} = -\frac{3}{2} \][/tex]
3. \(-\frac{3}{2}\) is a rational number (it can be expressed as a simple fraction).
### Option C: \(0.444444 \ldots\)
1. Express \(0.444444 \ldots\) as a fraction, which is \(\frac{4}{9}\) (since it is a repeating decimal).
2. When we multiply \(\frac{4}{9}\) by \(\frac{1}{4}\), we get:
[tex]\[ \frac{4}{9} \times \frac{1}{4} = \frac{4 \times 1}{9 \times 4} = \frac{4}{36} = \frac{1}{9} \][/tex]
3. \(\frac{1}{9}\) is a rational number (it can be expressed as a simple fraction).
### Option D: \(\frac{4}{3}\)
1. \(\frac{4}{3}\) is already in fractional form.
2. When we multiply \(\frac{4}{3}\) by \(\frac{1}{4}\), we get:
[tex]\[ \frac{4}{3} \times \frac{1}{4} = \frac{4 \times 1}{3 \times 4} = \frac{4}{12} = \frac{1}{3} \][/tex]
3. \(\frac{1}{3}\) is a rational number (it can be expressed as a simple fraction).
### Conclusion
Among the given options, the number that produces an irrational number when multiplied by \(\frac{1}{4}\) is \(\sqrt{12}\).
Thus, the correct answer is:
[tex]\[ \boxed{1 \text{ (A)}} \][/tex]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.