Answer :
To determine which number produces an irrational number when multiplied by 0.5, let's evaluate each given option step by step.
### Option A: [tex]\(\sqrt{16}\)[/tex]
First, we calculate the value of [tex]\(\sqrt{16}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
Now, multiply this result by 0.5:
[tex]\[ 0.5 \times 4 = 2.0 \][/tex]
Since 2.0 is a rational number, [tex]\(\sqrt{16}\)[/tex] does not produce an irrational number when multiplied by 0.5.
### Option B: [tex]\(0.555 \ldots\)[/tex]
Next, let's consider the value [tex]\(0.555 \ldots\)[/tex]:
[tex]\[ 0.5 \times 0.555 = 0.2775 \][/tex]
Since [tex]\(0.2775\)[/tex] is also a rational number, [tex]\(0.555 \ldots\)[/tex] does not produce an irrational number when multiplied by 0.5.
### Option C: [tex]\(\frac{1}{3}\)[/tex]
Evaluate [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{1}{3} \approx 0.333 \ldots \][/tex]
Now, multiply by 0.5:
[tex]\[ 0.5 \times \frac{1}{3} = \frac{1}{6} \approx 0.16666666666666666 \][/tex]
Since [tex]\(\frac{1}{6}\)[/tex] (or 0.16666666666666666) is a rational number, [tex]\(\frac{1}{3}\)[/tex] does not produce an irrational number when multiplied by 0.5.
### Option D: [tex]\(\sqrt{3}\)[/tex]
Finally, consider [tex]\(\sqrt{3}\)[/tex]. Since [tex]\(\sqrt{3}\)[/tex] is an irrational number,
Now, multiply it by 0.5:
[tex]\[ 0.5 \times \sqrt{3} \approx 0.8660254037844386 \][/tex]
Since [tex]\(0.8660254037844386\)[/tex] is an irrational number, [tex]\(\sqrt{3}\)[/tex] retains its irrationality when multiplied by 0.5.
### Conclusion
Among all given options, only [tex]\(\sqrt{3}\)[/tex] (Option D) produces an irrational number when multiplied by 0.5. Thus, the correct answer is:
[tex]\[ \boxed{\sqrt{3}} \][/tex]
### Option A: [tex]\(\sqrt{16}\)[/tex]
First, we calculate the value of [tex]\(\sqrt{16}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
Now, multiply this result by 0.5:
[tex]\[ 0.5 \times 4 = 2.0 \][/tex]
Since 2.0 is a rational number, [tex]\(\sqrt{16}\)[/tex] does not produce an irrational number when multiplied by 0.5.
### Option B: [tex]\(0.555 \ldots\)[/tex]
Next, let's consider the value [tex]\(0.555 \ldots\)[/tex]:
[tex]\[ 0.5 \times 0.555 = 0.2775 \][/tex]
Since [tex]\(0.2775\)[/tex] is also a rational number, [tex]\(0.555 \ldots\)[/tex] does not produce an irrational number when multiplied by 0.5.
### Option C: [tex]\(\frac{1}{3}\)[/tex]
Evaluate [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{1}{3} \approx 0.333 \ldots \][/tex]
Now, multiply by 0.5:
[tex]\[ 0.5 \times \frac{1}{3} = \frac{1}{6} \approx 0.16666666666666666 \][/tex]
Since [tex]\(\frac{1}{6}\)[/tex] (or 0.16666666666666666) is a rational number, [tex]\(\frac{1}{3}\)[/tex] does not produce an irrational number when multiplied by 0.5.
### Option D: [tex]\(\sqrt{3}\)[/tex]
Finally, consider [tex]\(\sqrt{3}\)[/tex]. Since [tex]\(\sqrt{3}\)[/tex] is an irrational number,
Now, multiply it by 0.5:
[tex]\[ 0.5 \times \sqrt{3} \approx 0.8660254037844386 \][/tex]
Since [tex]\(0.8660254037844386\)[/tex] is an irrational number, [tex]\(\sqrt{3}\)[/tex] retains its irrationality when multiplied by 0.5.
### Conclusion
Among all given options, only [tex]\(\sqrt{3}\)[/tex] (Option D) produces an irrational number when multiplied by 0.5. Thus, the correct answer is:
[tex]\[ \boxed{\sqrt{3}} \][/tex]
