Radicals & Exponents

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Answer Key & Explanations

Radicals & Exponents · Grade 9 · Worksheet 1

1. Isabella is designing a triangular garden with sides that form a right triangle. The two shorter sides measure 7√2 meters and 12√2 meters. What is the exact length of the hypotenuse in simplest radical form? Answer: √338 Solution: Use the Pythagorean theorem: a² + b² = c² Substitute the given side lengths: (7√2)² + (12√2)² = c² Calculate each square: (7√2)² = 7² × (√2)² = 49 × 2 = 98 Calculate the second square: (12√2)² = 12² × (√2)² = 144 × 2 = 288 Add the results: 98 + 288 = 386 So c² = 386, which means c = √386…
   Full step-by-step solution

   Step 1: Use the Pythagorean theorem: a² + b² = c² Step 2: Substitute the given side lengths: (7√2)² + (12√2)² = c² Step 3: Calculate each square: (7√2)² = 7² × (√2)² = 49 × 2 = 98 Step 4: Calculate the second square: (12√2)² = 12² × (√2)² = 144 × 2 = 288 Step 5: Add the results: 98 + 288 = 386 Step 6: So c² = 386, which means c = √386 Step 7: Simplify √386 by factoring: 386 = 2 × 193, and since 193 is prime, the radical cannot be simplified further Step 8: The exact length of the hypotenuse is √386 meters

2. √16 × √81 = ? Answer: 36 Solution: Calculate √16. Since 4 × 4 = 16, √16 = 4. Calculate √81.
   Full step-by-step solution

   Step 1: Calculate √16. Since 4 × 4 = 16, √16 = 4. Step 2: Calculate √81. Since 9 × 9 = 81, √81 = 9. Step 3: Multiply the results: 4 × 9 = 36. The answer is 36.

3. Aroha is designing a rectangular garden with an area of 98 square meters. The length of the garden is 7√2 meters. What is the width of the garden in simplest radical form? Answer: 7√2 Solution: The area of a rectangle is length × width. We know area = 98 and length = 7√2, so 7√2 × width = 98. Solve for width: width = 98 ÷ (7√2).
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width. Step 2: We know area = 98 and length = 7√2, so 7√2 × width = 98. Step 3: Solve for width: width = 98 ÷ (7√2). Step 4: Simplify: width = 14 ÷ √2. Step 5: Rationalize the denominator: width = (14 × √2) ÷ (√2 × √2) = (14√2) ÷ 2. Step 6: Simplify: width = 7√2. The width of the garden is 7√2 meters.

4. Aroha is building a rectangular garden with an area of 98 square meters. The length of the garden is √2 times its width. What is the width of Aroha's garden in simplest radical form? Answer: 7 Solution: Let w be the width of the garden. Since the length is √2 times the width, the length is w√2. The area of a rectangle is length × width, so w × w√2 = 98.
   Full step-by-step solution

   Step 1: Let w be the width of the garden. Since the length is √2 times the width, the length is w√2. Step 2: The area of a rectangle is length × width, so w × w√2 = 98. Step 3: This simplifies to w²√2 = 98. Step 4: Divide both sides by √2: w² = 98/√2. Step 5: Rationalize the denominator: w² = (98√2)/(√2×√2) = (98√2)/2 = 49√2. Step 6: Take the square root of both sides: w = √(49√2) = √49 × √(√2) = 7 × 2^(1/4). Step 7: Since the problem asks for simplest radical form, we need to express 2^(1/4) as a radical: w = 7√(√2) = 7 × the fourth root of 2. Step 8: However, looking back at step 5, we have w² = 49√2, which means w = √(49√2) = 7√(√2). This is the simplest radical form. The width is 7 meters.

5. √(45) × √(20) = ? Answer: 30 Solution: Multiply the radicands: √(45) × √(20) = √(45 × 20) Calculate 45 × 20 = 900 Simplify √(900) = 30 The answer is 30.
   Full step-by-step solution

   Step 1: Multiply the radicands: √(45) × √(20) = √(45 × 20) Step 2: Calculate 45 × 20 = 900 Step 3: Simplify √(900) = 30 Step 4: The answer is 30.

6. Isabella is designing a square garden with an area of 72 square meters. She needs to find the exact length of one side of the garden. What is the length in simplest radical form? Answer: 6√2 Solution: The area of a square is side × side, so side = √(area) Side = √72 Factor 72 into perfect squares: 72 = 36 × 2 √72 = √(36 × 2) = √36 × √2 √36 = 6, so 6 × √2 The simplified form is 6√2 Therefore, the length of one side is 6√2 meters
   Full step-by-step solution

   Step 1: The area of a square is side × side, so side = √(area) Step 2: Side = √72 Step 3: Factor 72 into perfect squares: 72 = 36 × 2 Step 4: √72 = √(36 × 2) = √36 × √2 Step 5: √36 = 6, so 6 × √2 Step 6: The simplified form is 6√2 Step 7: Therefore, the length of one side is 6√2 meters

7. Mere is designing a square mosaic for her art project. The area of the mosaic is 288 square centimeters. She needs to find the exact length of one side of the square mosaic. What is the exact side length in centimeters? Answer: 12√2 Solution: For a square, area = side × side, so side = √(area) Side length = √(288) Factor 288 into perfect squares: 288 = 144 × 2 √(288) = √(144 × 2) = √144 × √2 √144 = 12 Therefore, the exact side length is 12√2 centimeters
   Full step-by-step solution

   Step 1: For a square, area = side × side, so side = √(area) Step 2: Side length = √(288) Step 3: Factor 288 into perfect squares: 288 = 144 × 2 Step 4: √(288) = √(144 × 2) = √144 × √2 Step 5: √144 = 12 Step 6: Therefore, the exact side length is 12√2 centimeters

8. ∛(192) + 3∛(24) = ? Answer: 10∛3 Solution: Simplify ∛(192) 192 = 64 × 3 = 4³ × 3 ∛(192) = ∛(64 × 3) = ∛64 × ∛3 = 4∛3 Simplify 3∛(24) 24 = 8 × 3 = 2³ × 3 ∛(24) = ∛(8 × 3) = ∛8 × ∛3 = 2∛3 3∛(24) = 3 × 2∛3 = 6∛3 4∛3 + 6∛3 = (4 + 6)∛3 = 10∛3 The answer is 10∛3.
   Full step-by-step solution

   Step 1: Simplify ∛(192) 192 = 64 × 3 = 4³ × 3 ∛(192) = ∛(64 × 3) = ∛64 × ∛3 = 4∛3 Step 2: Simplify 3∛(24) 24 = 8 × 3 = 2³ × 3 ∛(24) = ∛(8 × 3) = ∛8 × ∛3 = 2∛3 3∛(24) = 3 × 2∛3 = 6∛3 Step 3: Combine like terms 4∛3 + 6∛3 = (4 + 6)∛3 = 10∛3 The answer is 10∛3.

9. ∛(81) + 2∛(3) = ? Answer: 5∛(3) Solution: Simplify ∛(81) 81 = 27 × 3 = 3³ × 3 ∛(81) = ∛(27 × 3) = ∛(27) × ∛(3) = 3∛(3) The expression becomes 3∛(3) + 2∛(3) Combine like terms (both terms have ∛(3)) 3∛(3) + 2∛(3) = (3 + 2)∛(3) = 5∛(3) The answer is 5∛(3).
   Full step-by-step solution

   Step 1: Simplify ∛(81) 81 = 27 × 3 = 3³ × 3 ∛(81) = ∛(27 × 3) = ∛(27) × ∛(3) = 3∛(3) Step 2: The expression becomes 3∛(3) + 2∛(3) Step 3: Combine like terms (both terms have ∛(3)) 3∛(3) + 2∛(3) = (3 + 2)∛(3) = 5∛(3) The answer is 5∛(3).