Question

1. make sense and persevere explain why constant terms expressed as different rational number types can be combined.

Explanation:

To explain why constant terms as different rational number types (e.g., integers, fractions, decimals) can be combined, we use the concept of equivalent forms. All rational numbers can be expressed as fractions (e.g., \( 5=\frac{5}{1} \), \( 0.25 = \frac{1}{4} \)). When combining constant terms, we find a common denominator (for fractions) or convert to a common form (e.g., all to decimals or all to fractions). For example, combining \( 3 \) (an integer, \( \frac{3}{1} \)) and \( \frac{1}{2} \): we get a common denominator (\( \frac{6}{2}+\frac{1}{2}=\frac{7}{2} \)). This works because rational numbers are closed under addition/subtraction (combining them gives another rational number), and the different forms are just representations of the same underlying rational number structure, so we can manipulate them to a common form for combination.

Answer:

Constant terms as different rational number types (e.g., integers, fractions, decimals) can be combined because all rational numbers can be expressed as fractions (e.g., \( a=\frac{a}{1} \) for integer \( a \), \( 0.5=\frac{1}{2} \) for a decimal). We convert them to a common form (e.g., common - denominator fractions or decimals), then combine using addition/subtraction rules. Rational numbers are closed under these operations, and different forms are equivalent representations of rational numbers, allowing manipulation to a common form for combination.