We will learn how to apply multiplication formulas to expressions containing radicals and expand and calculate them.
Applying Multiplication Formulas
Multiplication formulas using variables can be used in the same way with numbers containing radicals.
(a+b)² = a² + 2ab + b² (a+b)(a−b) = a² − b² and so on
📘 Example ①
Expand (√3 + 1)².
Solution: (√3)² + 2×√3×1 + 1² = 3 + 2√3 + 1 = 4 + 2√3
Expand (√3 + 1)².
Solution: (√3)² + 2×√3×1 + 1² = 3 + 2√3 + 1 = 4 + 2√3
📘 Example ②
Calculate (√5 + √2)(√5 − √2).
Solution: (√5)² − (√2)² = 5 − 2 = 3
Calculate (√5 + √2)(√5 − √2).
Solution: (√5)² − (√2)² = 5 − 2 = 3
💡 Key Points
- Remember that (√a)² = a
- An expression of the form (a+b)(a−b) results in a rational number
- Organize the radical part and integer part separately
Practice Problems
- Expand (√2 + 3)².
- Calculate (√6 + √2)(√6 − √2).
Answers and Explanations
- Answer: (√2)² + 2×√2×3 + 9 = 2 + 6√2 + 9 = 11 + 6√2
- Answer: (√6)² − (√2)² = 6 − 2 = 4