Question: Classify each number as rational or irrational and explain your choice for the numbers: $5 \frac{1}{25}$, $\sqrt{126}$, $0.15893...$, $-17$, and $0.7\overline{7}$. Write two examples of perfect squares and two examples of non-perfect squares.
1. **Classify each number as rational or irrational:**
1. $5 \frac{1}{25}$
- This is a mixed number which can be converted to an improper fraction: $5 \frac{1}{25} = \frac{126}{25}$.
- Since it can be expressed as a ratio of two integers, it is **rational**.
2. $\sqrt{126}$
- $126$ is not a perfect square.
- The square root of a non-perfect square is an **irrational** number.
3. $0.15893...$
- The ellipsis indicates the decimal continues without a repeating pattern.
- Non-repeating, non-terminating decimals are **irrational**.
4. $-17$
- $-17$ is an integer.
- All integers are rational because they can be written as $\frac{-17}{1}$.
- So, it is **rational**.
5. $0.7\overline{7}$
- This is a repeating decimal.
- All repeating decimals are **rational**.
2. **Examples of perfect squares:**
- $16 = 4^2$
- $81 = 9^2$
3. **Examples of non-perfect squares:**
- $20$ (no integer squared equals 20)
- $50$ (no integer squared equals 50)
**Final answers:**
| Value | Rational or Irrational | Explanation |
|----------------|-----------------------|-----------------------------------------------|
| $5 \frac{1}{25}$ | Rational | Can be expressed as $\frac{126}{25}$ |
| $\sqrt{126}$ | Irrational | Square root of non-perfect square |
| $0.15893...$ | Irrational | Non-repeating, non-terminating decimal |
| $-17$ | Rational | Integer, can be written as $\frac{-17}{1}$ |
| $0.7\overline{7}$ | Rational | Repeating decimal |