To determine whether \( 4.125 + \sqrt{3} \) will have an irrational answer, let's analyze the properties of the numbers involved:
1. Rational Numbers:
- A rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers \( p \) and \( q \), where \( q \neq 0 \).
- Examples of rational numbers include \( \frac{1}{2} \), 3, \( -5 \), and 4.125 since it can be written as \( \frac{33}{8} \).
2. Irrational Numbers:
- An irrational number is a number that cannot be expressed as a simple fraction. Its decimal representation goes on forever without repeating.
- Common examples include \( \pi \), \( e \), and \( \sqrt{2} \).
- Specifically, \( \sqrt{3} \) is an irrational number since its decimal representation is non-repeating and infinite.
Now let’s examine the expression \( 4.125 + \sqrt{3} \):
- 4.125 is a rational number. As previously mentioned, it can be expressed as the fraction \( \frac{33}{8} \), making it a rational number.
- \( \sqrt{3} \), on the other hand, is an irrational number. Its decimal expansion does not terminate or repeat.
According to the properties of rational and irrational numbers:
- When you add a rational number to an irrational number, the result is always an irrational number. This is because if the sum were rational, you would be able to express the irrational part as the difference between two rational numbers, which would be a contradiction as that difference would need to be irrational.
Therefore, adding the rational number 4.125 to the irrational number \( \sqrt{3} \) results in:
[tex]\[ 4.125 + \sqrt{3} \][/tex]
Since the result cannot be expressed as a ratio of two integers and maintains the property of non-repeating decimal expansion, the answer remains irrational.
Thus, the expression [tex]\( 4.125 + \sqrt{3} \)[/tex] has an irrational answer.
