Add and Subtract Radical Expressions - Algebra-2

1. Fundamental Concepts

Radical Expression: An expression of the form , where n is a positive integer (called the index) and a is the radicand. In addition and subtraction operations, square roots (, written as ) are commonly involved.
Like Radicals: Radicals that have the same radicand and the same index. For example, and are like radicals, while and are not (different radicands) and and are not (different indices).
Simplified Radical Form: A radical that meets the following two conditions:
- The radicand contains no factors or factors that can be taken out of the radical (i.e., no perfect square factors for square roots).
- The radicand contains no fractions. Before adding or subtracting radicals, all radicals must be converted to their simplified form first, and then check if they are like radicals.

2. Key Concepts

Prerequisite for Addition and Subtraction: Only Like Radicals Can Be Combined Only like radicals can be added or subtracted; unlike radicals cannot be directly combined (similar to "unlike fruits cannot be added together"). For example, cannot be further simplified.
Basis for Operation: Distributive Property of Multiplication When combining like radicals, keep the radicand unchanged and add or subtract the coefficients in front of the radicals (i.e., the "coefficients of the radicals"). This can be expressed by the formulas:； (Where m and n are rational numbers, , and a is a simplified radicand)
Operation Steps: "Simplify First → Identify Like Radicals → Combine Finally"
1. Simplify all radicals involved in the operation to ensure each radical is in its simplified form.
2. Identify and group like radicals.
3. Add or subtract the coefficients of each group of like radicals while keeping the radicand unchanged.
4. Retain unlike radicals as part of the result (if any).

3. Examples

1. Easy Difficulty (Directly Combine Simplified Like Radicals)

Question: Simplify
Solution:
Step 1: Confirm all radicals are like radicals (the radicand of each is 7, and they are already simplified).
Step 2: Combine the coefficients: .
Step 3: The result is .

2. Medium Difficulty (Simplify First, Then Combine Like Radicals)

Question: Simplify
Solution:
Step 1: Simplify each radical:

, so ;

;
Step 2: At this point, all radicals are like radicals (, , ).
Step 3: Combine the coefficients: .
Step 4: The result is .

3. Difficult Difficulty (Two Indices, Simplify First Then Calculate)

Question: Simplify （）
Solution: Step 1: Simplify radicals by index (3 for cube roots, 2 for square roots):

Cube roots (index=3):
- （，a perfect cube）；
- （，a perfect cube）。
Square roots (index=2):
- （，a perfect square）；
- （，a perfect square）。

Step 2: Group and combine like radicals:

Cube root group: ；
Square root group (simplified to linear terms): 。

Step 3: Final result: （unlike radicals, cannot be merged further）。

4. Problem-Solving Techniques

Prioritize Simplification to Avoid "Omission of Simplification" Errors When encountering non - simplified radicals, you must first factor the radicand (decompose it into "a perfect square number × a non - perfect square number") and then extract the square root of the perfect square number. For example, should first be converted to before determining if it is like any other radical.
Add/Subtract Coefficients Without Altering the Radical Sign When combining like radicals, only perform operations on the coefficients in front of the radical signs, and the radicand remains unchanged at all times. A common mistake is to add the radicands (e.g., ; the correct result is ).
Group and Combine in Multi - Step Operations If an expression contains multiple radicals (e.g., ), you can first group the like radicals (such as ) and then combine them separately to reduce calculation errors.