Question

Determine which of these expressions is a rational expression.(5 points) rational expression
not a rational expression
Z+1/z-8
2+ squared 8
a^2+5a+6/a+3
3 pie+7
2/9x

Answer 1

To determine which of the given expressions are rational expressions, we need to recall that a rational expression is defined as a fraction where both the numerator and the denominator are polynomials. Let's evaluate each of the expressions you provided:

1. **Z + 1/z - 8**: This is not a rational expression because it involves division of a variable (1/z) which is not a polynomial.

2. **2 + squared 8**: This expression does not clearly indicate a rational expression. If it refers to \(2 + 8\) (since squared 8 refers to \(sqrt{8}\)), then it's a constant and not a fraction. If it refers to \(2 + \sqrt{8}\), it is also not a fraction, thus not a rational expression.

3. **a^2 + 5a + 6/a + 3**: This seems to be a rational expression, but it's ambiguous due to lack of parentheses. If it is intended to be \((a^2 + 5a + 6)/(a + 3)\), then it is a rational expression because both the numerator and the denominator are polynomials. If it is improperly formatted as \(a^2 + 5a + 6/a + 3\), then it would not be clearer without parentheses.

4. **3 pie + 7**: Assuming 'pie' is just a misspelling for 'pi', \(3\pi + 7\) is a constant and not a fraction, thus not a rational expression.

5. **2/9x**: This is a rational expression since it is in the form of a fraction with \(2\) (a constant) over \(9x\) (a linear polynomial).

Based on this analysis, the expressions that qualify as rational expressions are:

- \( \frac{a^2 + 5a + 6}{a + 3} \) (assuming the correct formatting)
- \( \frac{2}{9x} \)

Final Answer: The rational expressions are **\( \frac{a^2 + 5a + 6}{a + 3} \)** and **\( \frac{2}{9x} \)**.