Multiply Radical Expressions

1. Fundamental Concepts

To master the multiplication of radical expressions, it is essential to first understand the following core terms, which form the basis for comprehending subsequent rules:
- Radical Symbol: The symbol "$\sqrt{}$" used to denote a root operation. It is a distinctive element of radical expressions and encloses the radicand.
- Index: The number located at the top-left corner of the radical symbol (if not explicitly shown, the index defaults to 2, representing a square root). It indicates "the degree of the root". For example, in $\sqrt[3]{a}$, the index is 3, meaning the cube root of a is being calculated.
- Radicand: The number or algebraic expression inside the radical symbol, which is the object of the root operation. For instance, in $\sqrt{x+y}$, "$x+y$" is the radicand.
- Prerequisite Condition: According to the definition of radical expressions, to ensure the operation is valid within the real number system, the radicands must satisfy $a\geq0$ and $b\geq0$ (where a and b are the radicands of the two radical expressions) when performing multiplication.

2. Key Concepts

This is the main form of radical multiplication. The rule states: When multiplying two or more radical expressions with the same index, the index remains unchanged. The product of the radicands serves as the new radicand, and the product of the coefficients outside the radicals becomes the new coefficient.
Expressed in formula form:
$\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b}$ (where $a\geq0$, $b\geq0$, and n is a positive integer)
If the radicals have coefficients (constants or algebraic expressions outside the radical symbol), the formula is extended to:
$m\sqrt[n]{a} \times k\sqrt[n]{b} = (m \times k) \times \sqrt[n]{a \times b}$ (where m and k are constants or integral expressions, and $a\geq0$, $b\geq0$)

3. Examples

(1) Easy

Question 1: Calculate $\sqrt{4} \times \sqrt{9}$
Solution:

Confirm the index: Both radicals are square roots (index = 2, same), satisfying the multiplication condition.
Apply the multiplication property: $\sqrt{4} \times \sqrt{9} = \sqrt{4 \times 9} = \sqrt{36}$
Simplify the result: $\sqrt{36} = 6$
Final Answer: 6

Question 2: Calculate $\sqrt[3]{8} \times \sqrt[3]{-27}$
Solution:

Confirm conditions: Index = 3 (odd, so radicands can be negative); same index, so direct multiplication is allowed.
Apply the multiplication property: $\sqrt[3]{8 \times (-27)} = \sqrt[3]{-216}$
Simplify: $\sqrt[3]{-216} = -6$ (since $(-6)^3 = -216$)
Final Answer: $-6$

(2) Medium

Question: Calculate $2\sqrt[3]{4x} \times 5\sqrt[3]{2x^2}$ ($x\in\mathbb{R}$)
Solution:

Confirm conditions: Index = 3 (odd, so x can be any real number); same index.
Separate coefficients and radicals: $(2 \times 5) \times \sqrt[3]{4x \times 2x^2} = 10\sqrt[3]{8x^3}$
Simplify the radical: $\sqrt[3]{8x^3} = \sqrt[3]{8} \times \sqrt[3]{x^3} = 2x$
Combine results: $10 \times 2x = 20x$
Final Answer: 20x
(3) Difficult

Question: Calculate $\sqrt{x(x+3)} \times \sqrt{(x+3)^3}$ ($x\geq0$, ensuring the radicands are non-negative)
Solution:
1. Confirm the conditions: Both indices are 2. The radicands $x(x+3)\geq0$ (holds when $x\geq0$) and $(x+3)^3\geq0$ (since a square is non-negative, and multiplying by $x+3$ (which is positive as $x\geq0$) keeps it non-negative, so this holds).
2. Apply the multiplication property: $\sqrt{x(x+3) \times (x+3)^3} = \sqrt{x \times (x+3)^{1+3}} = \sqrt{x \times (x+3)^4}$
3. Split the radicand: $\sqrt{(x+3)^4 \times x} = \sqrt{(x+3)^4} \times \sqrt{x}$
4. Simplify the radical of the power: $\sqrt{(x+3)^4} = (x+3)^2$ (since $(x+3)^2$ is non-negative, its square root is itself).
5. Combine the results: $(x+3)^2\sqrt{x}$ (can be expanded to $(x^2 + 6x + 9)\sqrt{x}$; both forms are correct, and the simplest form is $(x+3)^2\sqrt{x}$).
  Final Answer: $(x+3)^2\sqrt{x}$

4. Problem-Solving Techniques

(1) "Judge First, Calculate Later": Prioritize Confirming Operation Prerequisites

The first step in solving a problem is to check two key conditions:
- Whether the indices are the same: Only radicals with the same index can use the multiplication property directly; radicals with different indices need to be converted first (not covered in this section).
- Whether the radicands are non-negative: If the radicand contains a variable, the range of the variable (e.g., $x\geq0$) must be specified to avoid meaninglessness within the real number system.
(2) "Separate Operations": Split Coefficients and Radicals

If a radical has a coefficient (e.g., $m\sqrt[n]{a}$), follow the principle of "calculate coefficients separately and radicals separately": First multiply all coefficients, then apply the multiplication property to all radicals, and finally multiply the results of the two parts to avoid confusing coefficients with radicands.

(3) "Multiply First, Simplify Later": Simplify Step-by-Step to Reduce Errors

When calculating, first multiply the radicands as a whole, then simplify the result (instead of simplifying each radical individually). This reduces repetitive steps. When simplifying, pay attention to:
- Extracting "perfect squares" (for index 2) or "perfect n-th powers" (for index n) from the radicand. For example, $\sqrt{18x^2} = \sqrt{9x^2 \times 2} = 3x\sqrt{2}$.
- If the radicand is in the form of a power, use the rule "when multiplying powers with the same base, keep the base unchanged and add the exponents" (e.g., $(x+3)^1 \times (x+3)^3 = (x+3)^4$), then simplify using the relationship between radicals and powers (e.g., $\sqrt{(x+3)^4} = (x+3)^2$).
(4) "Verify the Result": Check if It Is in the Simplest Form

After completing the operation, confirm that the result meets the requirements of a "simplified radical":
- The radicand contains no factors or factors that can be taken out of the radical (i.e., no perfect powers).
- The radicand contains no denominators (denominator issues are mainly involved in division, which is only a supplementary reminder here). If these requirements are not met, further simplification is needed. For example, $6\sqrt{18x^2}$ needs to be simplified to $18x\sqrt{2}$ to be the final simplified form.

1. Fundamental Concepts

2. Key Concepts

3. Examples

(1) Easy

(2) Medium

(3) Difficult

4. Problem-Solving Techniques

(1) "Judge First, Calculate Later": Prioritize Confirming Operation Prerequisites

(2) "Separate Operations": Split Coefficients and Radicals

(3) "Multiply First, Simplify Later": Simplify Step-by-Step to Reduce Errors

(4) "Verify the Result": Check if It Is in the Simplest Form