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Let's go through the problems on this worksheet step-by-step. I’ll provide detailed solutions for each section.

### Section A: Absolute Value

1. **Evaluate the following absolute values**:
   a. \(|-18| = 18\)  
   b. \(|10 - 22| = | -12| = 12\)  
   c. \(|12 - 17| + | -9 + 5| = | -5| + | -4| = 5 + 4 = 9\)  
   d. \(| -8| + 5 × | -3| = 8 + 5 × 3 = 8 + 15 = 23\)  
   e. \(|2 - 7| + | -6 - 4| = | -5| + | -10| = 5 + 10 = 15\)  
   f. \(| -14| + 4 + 8 = 14 + 4 + 8 = 26\)

2. **Simplify the following expressions**:
   a. \(|-5 + 9| + |+11| = |4| + 11 = 4 + 11 = 15\)  
   b. \(4 × |6 - 15| + |10 - 6| = 4 × | -9| + |4| = 4 × 9 + 4 = 36 + 4 = 40\)  
   c. \(|-20 + 12| + |2 × ( -5)| = | -8| + | -10| = 8 + 10 = 18\)  
   d. \(|7 - 12| × | -9| = | -5| × 9 = 5 × 9 = 45\)

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### Section B: Rationalizing the Denominator

1. \(\frac{3}{\sqrt{2}} = \frac{3}{\sqrt{2}} × \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}\)

2. \(\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} × \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}\)

3. \(\frac{7}{\sqrt{5}} = \frac{7}{\sqrt{5}} × \frac{\sqrt{5}}{\sqrt{5}} = \frac{7\sqrt{5}}{5}\)

4. \(\frac{4}{\sqrt{6}} = \frac{4}{\sqrt{6}} × \frac{\sqrt{6}}{\sqrt{6}} = \frac{4\sqrt{6}}{6} = \frac{2\sqrt{6}}{3}\)

5. \(\frac{2}{3\sqrt{2}} = \frac{2}{3\sqrt{2}} × \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{3 × 2} = \frac{\sqrt{2}}{3}\)

6. \(\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} × \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}\)

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### Section C: Operations with Radicals (Square Roots Only)

1. \(\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}\)

2. \(\sqrt{50} - \sqrt{32} = 5\sqrt{2} - 4\sqrt{2} = \sqrt{2}\)

3. \(2\sqrt{12} + 3\sqrt{27} = 2 × 2\sqrt{3} + 3 × 3\sqrt{3} = 4\sqrt{3} + 9\sqrt{3} = 13\sqrt{3}\)

4. \(\sqrt{75} - 2\sqrt{48} = 5\sqrt{3} - 2 × 4\sqrt{3} = 5\sqrt{3} - 8\sqrt{3} = -3\sqrt{3}\)

5. \(3\sqrt{20} + \sqrt{45} - 2\sqrt{5} = 3 × 2\sqrt{5} + 3\sqrt{5} - 2\sqrt{5} = 6\sqrt{5} + 3\sqrt{5} - 2\sqrt{5} = 7\sqrt{5}\)

6. \(4\sqrt{18} - 3\sqrt{8} + 2\sqrt{50} = 4 × 3\sqrt{2} - 3 × 2\sqrt{2} + 2 × 5\sqrt{2} = 12\sqrt{2} - 6\sqrt{2} + 10\sqrt{2} = 16\sqrt{2}\)

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Would you like more details on any of these solutions?

### Related questions:

1. How do you calculate the absolute value of a number?
2. Why is rationalizing the denominator important in simplifying radicals?
3. Can all radicals be simplified in the same way?
4. What is the difference between adding radicals and multiplying them?
5. How can you check if an expression with radicals is fully simplified?

**Tip**: Always simplify the radical before performing operations to avoid unnecessary complexity.

Radicals and Absolute Value - Worksheet D for Grade 9 Mathematics

Explore detailed solutions to the Grade 9 Mathematics worksheet on radicals and absolute values, including absolute value calculations, rationalizing denominators, and operations with radicals. The step-by-step explanations provide insights into simplifying expressions and handling square roots.

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