To decide whether the expression [tex]\(\frac{\sqrt{3}}{3} \times \frac{18}{23}\)[/tex] represents a rational or an irrational number, let’s analyze each part of the product.
First, consider the number [tex]\(\frac{\sqrt{3}}{3}\)[/tex].
- [tex]\(\sqrt{3}\)[/tex] is known to be an irrational number because it cannot be expressed as a fraction [tex]\( \frac{a}{b} \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers and [tex]\( b \neq 0 \)[/tex].
- When an irrational number is divided by a rational number (in this case, 3), the result is still an irrational number.
Therefore, [tex]\(\frac{\sqrt{3}}{3}\)[/tex] is an irrational number.
Next, examine the number [tex]\(\frac{18}{23}\)[/tex]:
- [tex]\(\frac{18}{23}\)[/tex] is a fraction where both the numerator (18) and the denominator (23) are integers, and 23 is not equal to zero.
- Any number that can be expressed as a fraction of two integers is a rational number.
Thus, [tex]\(\frac{18}{23}\)[/tex] is a rational number.
To determine the nature of the product [tex]\(\frac{\sqrt{3}}{3} \times \frac{18}{23}\)[/tex], consider the properties of rational and irrational numbers:
- The product of an irrational number and a rational number is always irrational.
Combining these observations:
- [tex]\(\frac{\sqrt{3}}{3}\)[/tex] is irrational.
- [tex]\(\frac{18}{23}\)[/tex] is rational.
- The product of an irrational number and a rational number (in this case, [tex]\(\frac{\sqrt{3}}{3} \times \frac{18}{23}\)[/tex]) is irrational.
So, the product [tex]\(\frac{\sqrt{3}}{3} \times \frac{18}{23}\)[/tex] represents an irrational number.
This is because the number [tex]\(\frac{\sqrt{3}}{3}\)[/tex] is irrational and the number [tex]\(\frac{18}{23}\)[/tex] is rational. The product of an irrational number and a rational number is irrational.
