Divide Radical Expressions

4. Problem-Solving Techniques

• (1) "Index & Sign Check": Avoid Invalid Operations

  • First, confirm if the radicals have the same index: If yes, proceed with the division property; if no, convert to a common index (LCM of original indices) first.
  • For even indices: Strictly ensure the numerator’s radicand is non-negative and the denominator’s radicand is positive (never zero or negative).
  • For odd indices: Only ensure the denominator’s radicand is non-zero (numerator can be negative).

  (2) "Simplify First, Divide Later": Reduce Calculation Complexity

  Before applying the division property, simplify individual radicals if possible (e.g., factor out perfect n-th powers from radicands). This avoids dealing with large numbers later. Example:  $\frac{\sqrt{18x}}{\sqrt{2x}} = \frac{\sqrt{9 \times 2x}}{\sqrt{2x}} = \frac{3\sqrt{2x}}{\sqrt{2x}} = 3$  (simplifying first eliminates the need for division of radicands).

  (3) "Rationalization Step-by-Step": Match Methods to Denominator Type

  • Single radical denominator: Multiply numerator and denominator by  $\sqrt[n]{k^{n-1}}$  (where k is the radicand, n is the index) to make the denominator’s radicand a perfect n-th power.
  • Binomial radical denominator: Always use the conjugate (flip the sign between the two terms) to eliminate radicals—this works because the product of a binomial and its conjugate is a rational number (difference of squares).
  • After rationalizing, check if the resulting radical can be further simplified (e.g.,  $\sqrt[4]{32z^5}$  →  $2z\sqrt[4]{2z}$ ).

  (4) "Coefficient Handling": Keep Coefficients Separate

  When radicals have coefficients (e.g.,  $\frac{m\sqrt[n]{a}}{k\sqrt[n]{b}}$ ), calculate the quotient of the coefficients first (e.g.,  $\frac{m}{k}$ ), then handle the radical division separately. This avoids mixing coefficients with radicands and reduces errors. Example:  $\frac{6\sqrt{18x}}{3\sqrt{2x}} = \frac{6}{3} \times \sqrt{\frac{18x}{2x}} = 2 \times 3 = 6$  (coefficients calculated first, then radicals).

  (5) "Result Verification": Ensure Simplest Form

  After solving, confirm the final expression meets two criteria:
  
  
  1. No radicals remain in the denominator.
  2. The radicand contains no perfect n-th powers (matching the final index) and no common factors with the coefficient outside the radical. If either criterion is violated, further simplify (e.g.,  $\frac{4\sqrt[3]{24y^2}}{3y}$  →  $\frac{8\sqrt[3]{3y^2}}{3y}$ ).