To determine whether the expression [tex]\(\frac{2}{5} + \sqrt{3}\)[/tex] is rational or irrational, let's analyze the nature of the numbers involved.
1. Identifying the components of the expression:
- [tex]\(\frac{2}{5}\)[/tex] is a rational number because it can be expressed as the quotient of two integers (2 and 5).
- [tex]\(\sqrt{3}\)[/tex] is known to be an irrational number because it cannot be expressed as a quotient of two integers, and its decimal representation is non-repeating and non-terminating.
2. Understanding the properties of rational and irrational numbers:
- A rational number is one that can 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].
- An irrational number cannot be expressed as a simple fraction and has a non-terminating, non-repeating decimal expansion.
3. Combining a rational number and an irrational number:
- When you add a rational number to an irrational number, the result is always irrational. This is because the addition of a rational number to an irrational number cannot cause the non-repeating, non-terminating nature of the irrational number to disappear.
In our case:
- [tex]\(\frac{2}{5}\)[/tex] is rational.
- [tex]\(\sqrt{3}\)[/tex] is irrational.
- Therefore, [tex]\(\frac{2}{5} + \sqrt{3}\)[/tex] will be irrational.
Hence, the correct answer is:
d. The answer would be irrational. A rational number plus an irrational number is always irrational.
