Simplify the expression (3)/(2sqrt(6) - 3)

Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of 2sqrt(6) - sqrt(3) is 2sqrt(6) + sqrt(3). (3)/(2sqrt(6) - 3) × 2sqrt(6) + sqrt(3)2sqrt(6) + sqrt(3) Step 2: Multiply the numerators. 3(2sqrt(6) + sqrt(3)) = 6sqrt(6) + 3sqrt(3) Step 3: Multiply the denominators using the difference of squares formula (a-b)(a+b) = a^2 - b^2. Here, a = 2sqrt(6) and b = sqrt(3). (2sqrt(6))^2 - (sqrt(3))^2 (2^2 × (sqrt(6))^2) - 3 (4 × 6) - 3 24 - 3 = 21 Step 4: Combine the simplified numerator and denominator. 6sqrt(6) + 3sqrt(3)21 Step 5: Simplify the fraction by dividing all terms by the common factor, which is 3. 3(2sqrt(6) + sqrt(3))21 = 2sqrt(6) + sqrt(3)7 The simplified expression is 2sqrt(6) + sqrt(3)7. i) Set A set is a well-defined collection of distinct objects, called elements. For example, the set of even numbers less than 10 is \2, 4, 6, 8\. Diagram: A set can be represented by a circle or oval containing its elements. ` _______ / \ | A | | 1, 2, 3| \_______/ ` ii) Singleton Set A singleton set is a set that contains exactly one element. For example, the set of even prime numbers is \2\. Diagram: A singleton set is shown as a circle or oval containing only one element. ` _______ / \ | B | | 5 | \_______/ ` iii) Venn Diagram A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams use overlapping circles or other shapes to represent sets, with the universal set often represented by a rectangle enclosing all other sets. Diagram: This diagram shows a universal set (U) and two overlapping sets (A and B). The overlapping region represents elements common to both A and B (A B). ` _________________________ | U | | _______ _______ | | / \ / \| | | A | | B || | | | | || | \_______/ \_______/ | |_________________________| ` 3 done, 2 left today. You're making progress.