We will learn how to transform fractions with radicals in the denominator so that the denominator becomes an integer (rationalization), as well as how to perform mixed operations.
Rationalizing the Denominator
Multiply the numerator and denominator by the same radical to remove the radical from the denominator.
Example: 1 ÷ √a = √a ÷ a
📘 Example ①
Rationalize 1 ÷ √3.
Solution: 1/√3 = √3/(√3×√3) = √3/3
Rationalize 1 ÷ √3.
Solution: 1/√3 = √3/(√3×√3) = √3/3
📘 Example ② (Mixed Operations)
Calculate √3(√3 + √12).
Solution: Since √12 = 2√3, √3(1×√3 + 2√3) = √3 × 3√3 = 3×3 = 9
Calculate √3(√3 + √12).
Solution: Since √12 = 2√3, √3(1×√3 + 2√3) = √3 × 3√3 = 3×3 = 9
💡 Key Points
- When rationalizing, multiply the denominator and numerator by the same radical
- In mixed operations, perform multiplication and division first, followed by addition and subtraction
- Simplify the expressions inside the radicals before continuing the calculation
Practice Problems
- Rationalize 6 ÷ √2.
- Calculate √6 × √2 + √3.
Answers and Explanations
- Answer: 6/√2 = 6√2/2 = 3√2
- Answer: √6 × √2 = √12 = 2√3, so 2√3 + √3 = 3√3