Answer :
Certainly! Let's analyze each expression step-by-step to determine whether the result will be a rational or irrational number.
### Expression 1: \(\sqrt{5} + 7.28\)
1. Identify the components:
- \(\sqrt{5}\): Square root of 5 is an irrational number.
- \(7.28\): This is a rational number (since it can be expressed as a fraction).
2. Sum of a rational and an irrational:
- When you add a rational number to an irrational number, the result will always be irrational.
Thus, the expression \(\sqrt{5} + 7.28\) is irrational.
### Expression 2: \(-9 + \pi\)
1. Identify the components:
- \(-9\): This is a rational number.
- \(\pi\): Pi is an irrational number.
2. Sum of a rational and an irrational:
- Similar to the first expression, adding a rational number to an irrational number results in an irrational number.
Therefore, the expression \(-9 + \pi\) is irrational.
### Expression 3: \(23 \overline{.2} + \sqrt{16}\)
1. Identify the components:
- \(23 \overline{.2}\): This represents a repeating decimal (23.222...), which is a rational number.
- \(\sqrt{16}\): Square root of 16 is 4, which is a rational number.
2. Sum of two rational numbers:
- The sum of two rational numbers is always rational.
Hence, the expression \(23 \overline{.2} + \sqrt{16}\) is rational.
### Summary:
1. \(\sqrt{5} + 7.28\) is irrational.
2. \(-9 + \pi\) is irrational.
3. \(23 \overline{.2} + \sqrt{16}\) is rational.
So, the rationality/irrationality of each expression is as follows:
- \(\sqrt{5} + 7.28\,\) → Irrational
- \(-9 + \pi\,\) → Irrational
- [tex]\(23 \overline{.2} + \sqrt{16}\)[/tex] → Rational
### Expression 1: \(\sqrt{5} + 7.28\)
1. Identify the components:
- \(\sqrt{5}\): Square root of 5 is an irrational number.
- \(7.28\): This is a rational number (since it can be expressed as a fraction).
2. Sum of a rational and an irrational:
- When you add a rational number to an irrational number, the result will always be irrational.
Thus, the expression \(\sqrt{5} + 7.28\) is irrational.
### Expression 2: \(-9 + \pi\)
1. Identify the components:
- \(-9\): This is a rational number.
- \(\pi\): Pi is an irrational number.
2. Sum of a rational and an irrational:
- Similar to the first expression, adding a rational number to an irrational number results in an irrational number.
Therefore, the expression \(-9 + \pi\) is irrational.
### Expression 3: \(23 \overline{.2} + \sqrt{16}\)
1. Identify the components:
- \(23 \overline{.2}\): This represents a repeating decimal (23.222...), which is a rational number.
- \(\sqrt{16}\): Square root of 16 is 4, which is a rational number.
2. Sum of two rational numbers:
- The sum of two rational numbers is always rational.
Hence, the expression \(23 \overline{.2} + \sqrt{16}\) is rational.
### Summary:
1. \(\sqrt{5} + 7.28\) is irrational.
2. \(-9 + \pi\) is irrational.
3. \(23 \overline{.2} + \sqrt{16}\) is rational.
So, the rationality/irrationality of each expression is as follows:
- \(\sqrt{5} + 7.28\,\) → Irrational
- \(-9 + \pi\,\) → Irrational
- [tex]\(23 \overline{.2} + \sqrt{16}\)[/tex] → Rational
