Answer :
To determine which number produces an irrational number when multiplied by [tex]\(0.5\)[/tex], let's analyze each of the given choices step-by-step:
1. Choice A: [tex]\(\sqrt{16}\)[/tex]
[tex]\[ \sqrt{16} = 4 \][/tex]
When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ 4 \times 0.5 = 2 \][/tex]
Since 2 is a rational number, Choice A does not produce an irrational number.
2. Choice B: [tex]\(\sqrt{3}\)[/tex]
[tex]\[ \sqrt{3} \][/tex]
[tex]\(\sqrt{3}\)[/tex] is an irrational number by itself. When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ \sqrt{3} \times 0.5 = \frac{\sqrt{3}}{2} \][/tex]
Since the product of an irrational number and a nonzero rational number is still irrational, [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is irrational. Hence, Choice B produces an irrational number.
3. Choice C: [tex]\(\frac{1}{3}\)[/tex]
When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ \frac{1}{3} \times 0.5 = \frac{1}{6} \][/tex]
Since [tex]\(\frac{1}{6}\)[/tex] is a rational number, Choice C does not produce an irrational number.
4. Choice D: [tex]\(0.555\ldots\)[/tex] (repeating decimal)
This value is a rational number because repeating decimals can be expressed as fractions. For example:
[tex]\[ 0.555\ldots = \frac{5}{9} \][/tex]
When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ 0.555\ldots \times 0.5 = \frac{5}{9} \times 0.5 = \frac{5}{18} \][/tex]
Since [tex]\(\frac{5}{18}\)[/tex] is a rational number, Choice D does not produce an irrational number.
From the analysis, we see that the only choice that results in an irrational product when multiplied by [tex]\(0.5\)[/tex] is Choice B: [tex]\(\sqrt{3}\)[/tex].
Thus, the answer is:
[tex]\[ \boxed{B} \][/tex]
1. Choice A: [tex]\(\sqrt{16}\)[/tex]
[tex]\[ \sqrt{16} = 4 \][/tex]
When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ 4 \times 0.5 = 2 \][/tex]
Since 2 is a rational number, Choice A does not produce an irrational number.
2. Choice B: [tex]\(\sqrt{3}\)[/tex]
[tex]\[ \sqrt{3} \][/tex]
[tex]\(\sqrt{3}\)[/tex] is an irrational number by itself. When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ \sqrt{3} \times 0.5 = \frac{\sqrt{3}}{2} \][/tex]
Since the product of an irrational number and a nonzero rational number is still irrational, [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is irrational. Hence, Choice B produces an irrational number.
3. Choice C: [tex]\(\frac{1}{3}\)[/tex]
When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ \frac{1}{3} \times 0.5 = \frac{1}{6} \][/tex]
Since [tex]\(\frac{1}{6}\)[/tex] is a rational number, Choice C does not produce an irrational number.
4. Choice D: [tex]\(0.555\ldots\)[/tex] (repeating decimal)
This value is a rational number because repeating decimals can be expressed as fractions. For example:
[tex]\[ 0.555\ldots = \frac{5}{9} \][/tex]
When multiplied by [tex]\(0.5\)[/tex]:
[tex]\[ 0.555\ldots \times 0.5 = \frac{5}{9} \times 0.5 = \frac{5}{18} \][/tex]
Since [tex]\(\frac{5}{18}\)[/tex] is a rational number, Choice D does not produce an irrational number.
From the analysis, we see that the only choice that results in an irrational product when multiplied by [tex]\(0.5\)[/tex] is Choice B: [tex]\(\sqrt{3}\)[/tex].
Thus, the answer is:
[tex]\[ \boxed{B} \][/tex]
