Divide Binomial Radical Expressions - Algebra-2

1. Fundamental Concepts

Dividing Binomial Radical Expressions: A division operation where either the divisor or the dividend contains a binomial radical (e.g., \(a + b\sqrt{c}\), \(\sqrt{m} - \sqrt{n}\)). The core is to eliminate the radical in the denominator through "rationalizing the denominator" and convert it into a rational expression division.
Rationalizing the Denominator: The process of removing radicals from the denominator. For a denominator containing a binomial radical, this is achieved using a "conjugate" — multiply both the numerator and the denominator by the conjugate of the denominator to convert the denominator into a rational expression.
Conjugate in Division: Consistent with the definition of a conjugate in multiplying binomial radicals. For a denominator \((m + n\sqrt{p})\), its conjugate is \((m - n\sqrt{p})\), and vice versa. For example, the conjugate of the denominator \(5 - \sqrt{2}\) is \(5 + \sqrt{2}\), and the conjugate of the denominator \(\sqrt{3} + 2\sqrt{5}\) is \(\sqrt{3} - 2\sqrt{5}\).

2. Key Concepts

Essence of Division: Conversion to Multiplication of "Numerator × Denominator's Conjugate" Direct calculation of binomial radical division is not feasible. Instead, we multiply both the numerator and the denominator by the conjugate of the denominator. This transforms the denominator into the form \(a^2 - (b\sqrt{c})^2\) (using the difference of squares formula), which has no radicals. Then, expand and simplify the numerator to get the final result.
Key to Rationalizing the Denominator: Correct Selection and Application of Conjugates

If the denominator is in the form of "rational term + radical term" (e.g., \(3 + \sqrt{4}\)), the conjugate should have the sign between the two terms changed (e.g., \(3 - \sqrt{4}\)).
If the denominator is in the form of "radical term - radical term" (e.g., \(\sqrt{6} - \sqrt{2}\)), the conjugate should have the minus sign changed to a plus sign (e.g., \(\sqrt{6} + \sqrt{2}\)).
The numerator must be multiplied by the same conjugate as the denominator to keep the value of the fraction unchanged (based on the "fundamental property of fractions: multiplying both the numerator and denominator by a non-zero integral expression leaves the fraction's value unchanged").

Result Requires Simplification: Expand and Combine the Numerator + Simplify Radicals + Reduce the Fraction After multiplying the numerator by the conjugate, expand it using the FOIL method and combine like radicals. All radicals must be converted to their simplified form (e.g., \(\sqrt{12}\) → \(2\sqrt{3}\)). If there is a common factor between the numerator and the denominator, reduce the fraction to its simplest form (e.g., \(\frac{4\sqrt{2}}{2}\) → \(2\sqrt{2}\)).

3. Examples

1. Easy Difficulty (Denominator: "Rational Term + Single Radical Term")

Question: Calculate \(\frac{6}{2 + \sqrt{3}}\)
Solution:
Step 1: Identify the conjugate of the denominator: \(2 - \sqrt{3}\);
Step 2: Multiply both the numerator and denominator by the conjugate:\(\frac{6(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}\)
Step 3: Simplify the denominator (using the difference of squares formula):\((2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1\)
Step 4: Simplify the numerator and organize the result:\(6(2 - \sqrt{3}) = 12 - 6\sqrt{3}\). The final result is \(12 - 6\sqrt{3}\) (the denominator is 1 and can be omitted).

2. Medium Difficulty (Denominator: "Dual Radical Terms", Numerator Contains Radicals)

Question: Calculate \(\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}\)
Solution:
Step 1: Identify the conjugate of the denominator: \(\sqrt{5} + \sqrt{2}\);
Step 2: Multiply both the numerator and denominator by the conjugate:\(\frac{(\sqrt{5} + \sqrt{2})(\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})}\)
Step 3: Simplify the denominator and the numerator respectively:
Denominator: \((\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3\);
Numerator (using the perfect square formula): \((\sqrt{5})^2 + 2\sqrt{5}\times\sqrt{2} + (\sqrt{2})^2 = 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}\);
Step 4: Organize the result: \(\frac{7 + 2\sqrt{10}}{3}\) (there is no common factor between the numerator and the denominator, so no further reduction is needed).

3. Difficult Difficulty (Numerator and Denominator Contain Variables, Requiring Multi-Step Simplification)

Question: Calculate \(\frac{x - 4}{\sqrt{x} + 2}\) (where \(x \geq 0\) and \(x \neq 4\) to avoid the denominator being 0)
Solution:
Step 1: Observe the relationship between the denominator and the numerator: The numerator \(x - 4 = (\sqrt{x})^2 - 2^2\) (using the difference of squares formula), and the denominator is \(\sqrt{x} + 2\), whose conjugate is \(\sqrt{x} - 2\);
Step 2: Multiply both the numerator and denominator by the conjugate:\(\frac{(x - 4)(\sqrt{x} - 2)}{(\sqrt{x} + 2)(\sqrt{x} - 2)}\)
Step 3: Simplify the denominator and the numerator:
Denominator: \((\sqrt{x})^2 - 2^2 = x - 4\);
Numerator: \((x - 4)(\sqrt{x} - 2)\) (at this point, both the numerator and the denominator contain \(x - 4\), and since \(x \neq 4\), they can be canceled out);
Step 4: Cancel the common factor and organize the result: \(\frac{(x - 4)(\sqrt{x} - 2)}{x - 4} = \sqrt{x} - 2\).

4. Problem-Solving Techniques

Only Change the Sign of the Conjugate to Avoid Sign Errors When finding the conjugate of the denominator, only change the "+" or "-" between the two terms of the denominator; the sign of the coefficient in front of the radical remains unchanged. For example, the conjugate of the denominator \(3 - 2\sqrt{5}\) is \(3 + 2\sqrt{5}\) (only change "-" to "+", not the positive sign of "2").
First Observe the Numerator and Denominator, and Prioritize Using Formulas for Simplification If the numerator is in the form of a "difference of squares" (e.g., \(a^2 - b^2\)) and the denominator is \(a + b\) or \(a - b\), factor the numerator first (e.g., \(x - 9 = (\sqrt{x})^2 - 3^2\)), then cancel the common factor with the denominator to reduce subsequent expansion steps (as in the difficult difficulty example).
Expand the Numerator "Term by Term" and Combine Like Radicals Without Omission After multiplying the numerator by the conjugate, expand it term by term using the FOIL method (especially when there are multiple radicals or variables). Then combine like radicals (e.g., \(3\sqrt{6} + 5\sqrt{6} = 8\sqrt{6}\)) to avoid missing terms or calculation errors.
Final Check of "Three Key Points" to Ensure the Result is in Simplest Form After completing the calculation, verify three points: ① There are no radicals in the denominator; ② All radicals are in their simplified form; ③ There are no common factors between the numerator and the denominator (including integral common factors, e.g., \(\frac{2\sqrt{3} + 4}{2}\) can be simplified to \(\sqrt{3} + 2\)).