Identify rational algebraic expressions in this math worksheet.

Problem: Identify whether each expression is a rational algebraic expression or not.

A rational algebraic expression is defined as an expression that can be written in the form:

\[
\frac{P(x)}{Q(x)}
\]

where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \).

Let's analyze each expression step by step:

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#### 1. \( \frac{6}{4x} \)

- Numerator: \( 6 \) (a constant, which is a polynomial).
- Denominator: \( 4x \) (a polynomial).
- The expression is in the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
- Conclusion: This is a rational algebraic expression.

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#### 2. \( \frac{9}{n-n} \)

- Numerator: \( 9 \) (a constant, which is a polynomial).
- Denominator: \( n - n = 0 \).
- Division by zero is undefined, so this expression is not valid.
- Conclusion: This is not a rational algebraic expression.

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#### 3. \( \frac{2}{k^{\frac{1}{2}}} \)

- Numerator: \( 2 \) (a constant, which is a polynomial).
- Denominator: \( k^{\frac{1}{2}} \) (not a polynomial because it involves a fractional exponent).
- Since the denominator is not a polynomial, the expression is not in the form \( \frac{P(x)}{Q(x)} \).
- Conclusion: This is not a rational algebraic expression.

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#### 4. \( 81x + 1 \)

- This is a polynomial (specifically, a linear polynomial).
- A polynomial is not a rational algebraic expression unless it is written as a fraction with a non-zero polynomial denominator.
- Conclusion: This is not a rational algebraic expression.

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#### 5. \( \frac{x^{-2} + 30}{x - 30} \)

- Numerator: \( x^{-2} + 30 \). Here, \( x^{-2} \) is not a polynomial because it involves a negative exponent.
- Denominator: \( x - 30 \) (a polynomial).
- Since the numerator is not a polynomial, the expression is not in the form \( \frac{P(x)}{Q(x)} \).
- Conclusion: This is not a rational algebraic expression.

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#### 6. \( \frac{a^2b^3c^5}{abc} \)

- Numerator: \( a^2b^3c^5 \) (a polynomial in terms of \( a \), \( b \), and \( c \)).
- Denominator: \( abc \) (a polynomial in terms of \( a \), \( b \), and \( c \)).
- Both the numerator and the denominator are polynomials.
- Conclusion: This is a rational algebraic expression.

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#### 7. \( \sqrt{b - 5} \)

- This expression involves a square root, which is not a polynomial.
- Rational algebraic expressions must have polynomials in both the numerator and the denominator.
- Conclusion: This is not a rational algebraic expression.

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#### 8. \( \frac{r + \sqrt{3}}{4} \)

- Numerator: \( r + \sqrt{3} \). Here, \( \sqrt{3} \) is a constant, but the presence of \( r \) makes it a polynomial.
- Denominator: \( 4 \) (a constant, which is a polynomial).
- Both the numerator and the denominator are polynomials.
- Conclusion: This is a rational algebraic expression.

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#### 9. \( \left( \frac{6}{w} \right)^0 \)

- Any non-zero number raised to the power of 0 is 1.
- Therefore, \( \left( \frac{6}{w} \right)^0 = 1 \).
- \( 1 \) is a constant, which is a polynomial.
- However, it is not in the form \( \frac{P(x)}{Q(x)} \) with a non-trivial denominator.
- Conclusion: This is not a rational algebraic expression.

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#### 10. \( xy^{\frac{1}{2}} \)

- This expression involves \( y^{\frac{1}{2}} \), which is not a polynomial because it has a fractional exponent.
- Rational algebraic expressions must have polynomials in both the numerator and the denominator.
- Conclusion: This is not a rational algebraic expression.

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Final Answers:

1. Rational
2. Not rational
3. Not rational
4. Not rational
5. Not rational
6. Rational
7. Not rational
8. Rational
9. Not rational
10. Not rational

\[
\boxed{
\begin{array}{ll}
1. & \text{Rational} \\
2. & \text{Not rational} \\
3. & \text{Not rational} \\
4. & \text{Not rational} \\
5. & \text{Not rational} \\
6. & \text{Rational} \\
7. & \text{Not rational} \\
8. & \text{Rational} \\
9. & \text{Not rational} \\
10. & \text{Not rational} \\
\end{array}
}
\]