Which of the following is a rational expression?


A. ﻿left parenthesis x plus 1 right parenthesis to the power of 7 divided by 2 end exponent﻿


﻿B. 3 to the power of x plus 2 to the power of x plus 8﻿


C. ﻿square root of x squared plus 5 x plus 6 end root﻿


D. ﻿fraction numerator 2 over denominator x plus 1 end fraction﻿

Let's analyze each option to determine which one is a rational expression.

### Definition of a Rational Expression:
A rational expression is a fraction where both the numerator and the denominator are polynomials.

### Analysis of Each Option:

**A.** \(\frac{(x + 1)^7}{2}\)
- The numerator \((x + 1)^7\) is a polynomial.
- The denominator \(2\) is a constant, which is also considered a polynomial.
- **This is a rational expression.**

**B.** \(3^x + 2^x + 8\)
- This expression involves exponential terms \(3^x\) and \(2^x\).
- **This is not a rational expression** because it is not a fraction, and the terms are not polynomials.

**C.** \(\sqrt{x^2 + 5x + 6}\)
- The expression inside the square root is a polynomial, but taking the square root of it means it's not in polynomial form.
- **This is not a rational expression** because it involves a square root, which makes it irrational.

**D.** \(\frac{2}{x + 1}\)
- The numerator \(2\) is a polynomial.
- The denominator \(x + 1\) is also a polynomial.
- **This is a rational expression.**

### Conclusion:
Both options **A** and **D** are rational expressions, but since the question asks for "which of the following is **a** rational expression," and assuming only one answer is correct:

- **D** \(\frac{2}{x + 1}\) is the correct answer.

Would you like further explanation on this topic or have any other questions?

Here are some related questions you might find useful:
1. How do you simplify rational expressions?
2. What are the steps to identify non-rational expressions?
3. Can rational expressions include roots or exponents?
4. How do you add and subtract rational expressions?
5. What is the difference between rational and irrational expressions?

**Tip:** When identifying rational expressions, focus on whether the expression is a fraction of polynomials without roots or exponential terms.

Learn how to identify rational expressions by analyzing multiple choice options. This problem explores different mathematical expressions involving polynomials and fractions. Suitable for students in Grades 9-12.

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