Which number produces an irrational number when multiplied by [tex]$\frac{1}{3}$[/tex]?

A. [tex]$-\sqrt{16}$[/tex]

B. [tex]$0.777777 \ldots$[/tex]

C. [tex]$\sqrt{27}$[/tex]

D. [tex]$\frac{1}{3}$[/tex]



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]