Answer :
To determine which number produces an irrational number when multiplied by \(\frac{1}{3}\), let's analyze each option step-by-step.
Option A: \(-\sqrt{16}\)
First, simplify \(-\sqrt{16}\):
[tex]\[ -\sqrt{16} = -4 \][/tex]
Now, multiply by \(\frac{1}{3}\):
[tex]\[ -4 \times \frac{1}{3} = -\frac{4}{3} \][/tex]
\(-\frac{4}{3}\) is a rational number since it can be expressed as a fraction of two integers.
Option B: \(0.777777 \ldots\)
The number \(0.777777 \ldots\) is a repeating decimal, which can be written as:
[tex]\[ 0.\overline{7} = \frac{7}{9} \][/tex]
Now, multiply by \(\frac{1}{3}\):
[tex]\[ \frac{7}{9} \times \frac{1}{3} = \frac{7}{27} \][/tex]
\(\frac{7}{27}\) is a rational number since it can be expressed as a fraction of two integers.
Option C: \(\sqrt{27}\)
First, simplify \(\sqrt{27}\):
[tex]\[ \sqrt{27} = 3\sqrt{3} \][/tex]
Now, multiply by \(\frac{1}{3}\):
[tex]\[ 3\sqrt{3} \times \frac{1}{3} = \sqrt{3} \][/tex]
\(\sqrt{3}\) is an irrational number as it cannot be expressed as a fraction of two integers.
Option D: \(\frac{1}{3}\)
Now, multiply by \(\frac{1}{3}\):
[tex]\[ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \][/tex]
\(\frac{1}{9}\) is a rational number since it can be expressed as a fraction of two integers.
From this analysis, we see that multiplying \(\sqrt{27}\) by \(\frac{1}{3}\) gives the irrational number \(\sqrt{3}\). Therefore, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
Option A: \(-\sqrt{16}\)
First, simplify \(-\sqrt{16}\):
[tex]\[ -\sqrt{16} = -4 \][/tex]
Now, multiply by \(\frac{1}{3}\):
[tex]\[ -4 \times \frac{1}{3} = -\frac{4}{3} \][/tex]
\(-\frac{4}{3}\) is a rational number since it can be expressed as a fraction of two integers.
Option B: \(0.777777 \ldots\)
The number \(0.777777 \ldots\) is a repeating decimal, which can be written as:
[tex]\[ 0.\overline{7} = \frac{7}{9} \][/tex]
Now, multiply by \(\frac{1}{3}\):
[tex]\[ \frac{7}{9} \times \frac{1}{3} = \frac{7}{27} \][/tex]
\(\frac{7}{27}\) is a rational number since it can be expressed as a fraction of two integers.
Option C: \(\sqrt{27}\)
First, simplify \(\sqrt{27}\):
[tex]\[ \sqrt{27} = 3\sqrt{3} \][/tex]
Now, multiply by \(\frac{1}{3}\):
[tex]\[ 3\sqrt{3} \times \frac{1}{3} = \sqrt{3} \][/tex]
\(\sqrt{3}\) is an irrational number as it cannot be expressed as a fraction of two integers.
Option D: \(\frac{1}{3}\)
Now, multiply by \(\frac{1}{3}\):
[tex]\[ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \][/tex]
\(\frac{1}{9}\) is a rational number since it can be expressed as a fraction of two integers.
From this analysis, we see that multiplying \(\sqrt{27}\) by \(\frac{1}{3}\) gives the irrational number \(\sqrt{3}\). Therefore, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
