To determine which number produces an irrational number when multiplied by [tex]\(\frac{1}{4}\)[/tex], we need to analyze each given number and perform the multiplication, then check whether the result is irrational.
First, let's define what it means for a number to be irrational:
- A number is irrational if it cannot be expressed as a ratio of two integers, i.e., it has a non-repeating, non-terminating decimal expansion.
Given numbers:
A. [tex]\(\frac{4}{3}\)[/tex]
B. [tex]\(\sqrt{12}\)[/tex]
C. [tex]\(-\sqrt{36}\)[/tex]
D. [tex]\(0.444444 \ldots\)[/tex]
### Multiplications and Checks:
#### A. [tex]\(\frac{4}{3}\)[/tex]
[tex]\[
\frac{4}{3} \times \frac{1}{4} = \frac{4}{3} \cdot \frac{1}{4} = \frac{4 \cdot 1}{3 \cdot 4} = \frac{1}{3}
\][/tex]
[tex]\(\frac{1}{3}\)[/tex] is a rational number.
#### B. [tex]\(\sqrt{12}\)[/tex]
[tex]\[
\sqrt{12} \times \frac{1}{4} = \sqrt{12} \cdot \frac{1}{4} = \frac{\sqrt{12}}{4} = \frac{\sqrt{4 \cdot 3}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\(\frac{\sqrt{3}}{2}\)[/tex] is irrational because [tex]\(\sqrt{3}\)[/tex] is an irrational number and dividing an irrational number by a rational number still results in an irrational number.
#### C. [tex]\(-\sqrt{36}\)[/tex]
[tex]\[
-\sqrt{36} \times \frac{1}{4} = -\sqrt{36} \cdot \frac{1}{4} = \frac{-\sqrt{36}}{4} = \frac{-6}{4} = -\frac{3}{2}
\][/tex]
[tex]\(-\frac{3}{2}\)[/tex] is a rational number.
#### D. [tex]\(0.444444 \ldots\)[/tex]
[tex]\[
0.444444 \ldots \times \frac{1}{4} = 0.444444 \ldots \cdot \frac{1}{4} = 0.111111 \ldots
\][/tex]
[tex]\(0.111111 \ldots\)[/tex] (which is the repeating decimal [tex]\(0.\overline{1}\)[/tex]) is a rational number because it can be expressed as the fraction [tex]\(\frac{1}{9}\)[/tex].
### Conclusion:
The number that, when multiplied by [tex]\(\frac{1}{4}\)[/tex], produces an irrational number is [tex]\(\sqrt{12}\)[/tex] (option B).
