To determine whether the product of the two fractions
[tex]\[
\frac{8}{37} \times \frac{\sqrt{2}}{2}
\][/tex]
is rational or irrational, let's analyze each component step-by-step.
1. Analyze [tex]\(\frac{8}{37}\)[/tex]:
[tex]\(\frac{8}{37}\)[/tex] is a fraction where both the numerator (8) and the denominator (37) are integers. Since the denominator is not zero, this fraction is a rational number.
2. Analyze [tex]\(\frac{\sqrt{2}}{2}\)[/tex]:
[tex]\(\sqrt{2}\)[/tex] is the square root of 2, which is known to be an irrational number. An irrational number is a number that cannot be expressed as a simple fraction (a ratio of two integers).
Dividing this irrational number ([tex]\(\sqrt{2}\)[/tex]) by 2, which is a non-zero integer, still results in an irrational number. This is because dividing an irrational number by a rational number does not produce a rational number.
3. Analyze the product:
When you multiply a rational number by an irrational number:
[tex]\[
\frac{8}{37} \times \frac{\sqrt{2}}{2}
\][/tex]
In this case, [tex]\(\frac{8}{37}\)[/tex] is rational and [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is irrational. The product of a rational number and an irrational number will always be irrational. This is because multiplying does not change the inherent properties of irrational numbers—they cannot be expressed as fractions of integers, and combining them with rational numbers will not alter that fundamental property.
Therefore, the product
[tex]\[
\frac{8}{37} \times \frac{\sqrt{2}}{2}
\][/tex]
is an irrational number.
