To determine which number produces an irrational number when multiplied by [tex]\(\frac{2}{5}\)[/tex], let's analyze each option:
A. [tex]\(1.34\)[/tex] \\
[tex]\(1.34\)[/tex] is a rational number because it can be expressed as a fraction: [tex]\(\frac{134}{100}\)[/tex]. When a rational number (here, [tex]\(\frac{2}{5}\)[/tex]) is multiplied by another rational number, the result is always rational. Therefore, [tex]\(\frac{2}{5} \times 1.34\)[/tex] remains a rational number.
B. [tex]\(\frac{5}{7}\)[/tex] \\
[tex]\(\frac{5}{7}\)[/tex] is also a rational number. Similar to the previous case, multiplying two rational numbers yields a rational number. Hence, [tex]\(\frac{2}{5} \times \frac{5}{7}\)[/tex] results in a rational number.
C. [tex]\(\pi\)[/tex] \\
[tex]\(\pi\)[/tex] (pi) is an irrational number. An irrational number is one that cannot be expressed as a simple fraction. When a rational number (such as [tex]\(\frac{2}{5}\)[/tex]) is multiplied by an irrational number, the result is always irrational. So, [tex]\(\frac{2}{5} \times \pi\)[/tex] results in an irrational number.
D. [tex]\(-\frac{5}{2}\)[/tex] \\
[tex]\(-\frac{5}{2}\)[/tex] is a rational number. Multiplying a rational number by another rational number results in a rational number. Therefore, [tex]\(\frac{2}{5} \times -\frac{5}{2}\)[/tex] yields a rational number.
Out of the given choices, only option C (multiplying by [tex]\(\pi\)[/tex]) results in an irrational number.
Therefore, the number that produces an irrational number when multiplied by [tex]\(\frac{2}{5}\)[/tex] is:
[tex]\(\boxed{3}\)[/tex]
