To determine which number produces an irrational answer when added to [tex]\(\frac{3}{8}\)[/tex], we first need to understand the nature of rational and irrational numbers.
Rational numbers are numbers that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]. This includes all integers, finite decimals, and repeating decimals.
Irrational numbers are numbers that cannot be expressed as a simple fraction [tex]\(\frac{p}{q}\)[/tex]. Their decimal expansions are non-terminating and non-repeating.
Given numbers:
- [tex]\(\frac{2}{5}\)[/tex]
- [tex]\(1.732050807 \ldots\)[/tex]
- [tex]\(\sqrt{9}\)[/tex]
- 0.78
We need to analyze each number one by one:
1. [tex]\(\frac{2}{5}\)[/tex] is a rational number because it is a fraction of two integers. Adding [tex]\(\frac{2}{5}\)[/tex] to another rational number will yield a rational number:
[tex]\[
\frac{3}{8} + \frac{2}{5} = \frac{3 \cdot 5 + 8 \cdot 2}{8 \cdot 5} = \frac{15 + 16}{40} = \frac{31}{40}
\][/tex]
Since [tex]\(\frac{31}{40}\)[/tex] is a rational number, choice A does not produce an irrational number.
2. [tex]\(1.732050807 \ldots\)[/tex] is an approximation of [tex]\(\sqrt{3}\)[/tex], which is known to be an irrational number. The sum of a rational number ([tex]\(\frac{3}{8}\)[/tex]) and an irrational number ([tex]\(\sqrt{3}\)[/tex]) is always irrational:
[tex]\[
\frac{3}{8} + \sqrt{3}
\][/tex]
Since [tex]\(\sqrt{3}\)[/tex] is irrational and adding [tex]\(\sqrt{3}\)[/tex] to [tex]\(\frac{3}{8}\)[/tex] yields an irrational number, choice B does produce an irrational answer.
3. [tex]\(\sqrt{9}\)[/tex] evaluates to 3, which is a rational number (as [tex]\(3\)[/tex] is an integer). Adding 3 to another rational number ([tex]\(\frac{3}{8}\)[/tex]) yields a rational number:
[tex]\[
\frac{3}{8} + 3 = \frac{3}{8} + \frac{24}{8} = \frac{27}{8}
\][/tex]
Since [tex]\(\frac{27}{8}\)[/tex] is rational, choice C does not produce an irrational number.
4. [tex]\(\)[/tex] is 0.78 a terminating decimal and is hence a rational number. Adding 0.78 to another rational number ([tex]\(\frac{3}{8}\)[/tex]) yields a rational number:
[tex]\[
\frac{3}{8} + 0.78
\][/tex]
Converting both to decimals for simplicity,
[tex]\[
0.375 + 0.78 = 1.155
\][/tex]
Since 1.155 is a terminating decimal, it is rational. Therefore, choice D does not produce an irrational number.
Thus, the answer is:
B. [tex]\(1.732050807 \ldots\)[/tex] (which is an approximation of [tex]\(\sqrt{3}\)[/tex])
