Radicals & Exponents

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Answer Key & Explanations

Radicals & Exponents · Grade 9 · Worksheet 2

1. Matiu is designing a triangular garden with sides of length 2√8 meters, 3√2 meters, and √50 meters. What is the perimeter of the garden in simplest radical form? Answer: 12√2 Solution: 2√8 = 2√(4×2) = 2×2√2 = 4√2 3√2 = 3√2 (already simplified) √50 = √(25×2) = 5√2 Perimeter = 4√2 + 3√2 + 5√2 Perimeter = (4 + 3 + 5)√2 = 12√2 The perimeter of the garden is 12√2 meters.
   Full step-by-step solution

   Step 1: Simplify each radical term 2√8 = 2√(4×2) = 2×2√2 = 4√2 3√2 = 3√2 (already simplified) √50 = √(25×2) = 5√2 Step 2: Add all the simplified terms Perimeter = 4√2 + 3√2 + 5√2 Step 3: Combine like terms Perimeter = (4 + 3 + 5)√2 = 12√2 The perimeter of the garden is 12√2 meters.

2. Liam is building a square tabletop and needs to calculate the diagonal length. If each side of the square measures 5√20 inches, what is the length of the diagonal in simplest radical form? Answer: 10√10 Solution: The diagonal of a square with side length s is given by s√2 Substitute the given side length: diagonal = (5√20) × √2 Multiply the radicals: diagonal = 5√(20×2) = 5√40 Simplify √40: √40 = √(4×10) = √4 × √10 = 2√10 Multiply: 5 × 2√10 = 10√10 The answer is 10√10.
   Full step-by-step solution

   Step 1: The diagonal of a square with side length s is given by s√2 Step 2: Substitute the given side length: diagonal = (5√20) × √2 Step 3: Multiply the radicals: diagonal = 5√(20×2) = 5√40 Step 4: Simplify √40: √40 = √(4×10) = √4 × √10 = 2√10 Step 5: Multiply: 5 × 2√10 = 10√10 The answer is 10√10.

3. √(72) × √(18) = ? Answer: 36 Solution: Multiply the numbers under the radicals: √(72) × √(18) = √(72 × 18) Calculate 72 × 18 = 1296 Now we have √(1296) Factor 1296 to find perfect squares: 1296 = 36 × 36 √(36 × 36) = √36 × √36 = 6 × 6 = 36 Therefore, √(72) × √(18) = 36
   Full step-by-step solution

   Step 1: Multiply the numbers under the radicals: √(72) × √(18) = √(72 × 18) Step 2: Calculate 72 × 18 = 1296 Step 3: Now we have √(1296) Step 4: Factor 1296 to find perfect squares: 1296 = 36 × 36 Step 5: √(36 × 36) = √36 × √36 = 6 × 6 = 36 Step 6: Therefore, √(72) × √(18) = 36

4. Kaia is creating a square mosaic pattern for her art project. She needs to determine the side length of a square that has an area of 75 square centimeters. What is the simplified radical form of the side length? Answer: 5√3 Solution: The area of a square is given by side length squared, so side length = √(area).
   Full step-by-step solution

   Step 1: The area of a square is given by side length squared, so side length = √(area). Step 2: Substitute the given area: side length = √75 Step 3: Factor 75 into prime factors: 75 = 25 × 3 Step 4: Rewrite the square root: √75 = √(25 × 3) Step 5: Separate the square roots: √(25 × 3) = √25 × √3 Step 6: Simplify √25 to 5: 5 × √3 Step 7: The simplified radical form is 5√3 The answer is 5√3.

5. Matiu is designing a triangular garden with sides that form a right triangle. The longer leg measures 12 meters and the shorter leg measures 5 meters. What is the length of the hypotenuse in simplest radical form? Answer: 13 Solution: Use the Pythagorean theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse.
   Full step-by-step solution

   Step 1: Use the Pythagorean theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse. Step 2: Substitute the given values: 5² + 12² = c² Step 3: Calculate the squares: 25 + 144 = c² Step 4: Add the results: 169 = c² Step 5: Take the square root of both sides: c = √169 Step 6: Simplify the radical: √169 = 13 The answer is 13.

6. Mason is building a wooden frame for a triangular garden plot. The sides of the triangle measure √(128) feet, √(50) feet, and √(18) feet. He needs to calculate the total length of wood needed for the perimeter. What is the perimeter of the triangular garden plot in simplest radical form? Answer: A. 14√2 feet Solution: √(128) = √(64 × 2) = √(64) × √(2) = 8√2 √(50) = √(25 × 2) = √(25) × √(2) = 5√2 √(18) = √(9 × 2) = √(9) × √(2) = 3√2 Add the simplified expressions to find the perimeter Perimeter = 8√2 + 5√2 + 3√2 = (8 + 5 + 3)√2 = 16√2 The perimeter is 16√2 feet, which corresponds to option A.
   Full step-by-step solution

   Step 1: Simplify each radical expression √(128) = √(64 × 2) = √(64) × √(2) = 8√2 √(50) = √(25 × 2) = √(25) × √(2) = 5√2 √(18) = √(9 × 2) = √(9) × √(2) = 3√2 Step 2: Add the simplified expressions to find the perimeter Perimeter = 8√2 + 5√2 + 3√2 = (8 + 5 + 3)√2 = 16√2 Step 3: Compare with answer choices The perimeter is 16√2 feet, which corresponds to option A.

7. √16 × √36 = ? Answer: 24 Solution: Simplify √16 = 4 Simplify √36 = 6 Multiply the results: 4 × 6 = 24 The answer is 24.
   Full step-by-step solution

   Step 1: Simplify √16 = 4 Step 2: Simplify √36 = 6 Step 3: Multiply the results: 4 × 6 = 24 The answer is 24.

8. ∛(135) = ? Answer: 3∛5 Solution: Factor 135 into prime factors: 135 = 27 × 5 Since 27 is a perfect cube (3³ = 27), we can write: ∛(135) = ∛(27 × 5) Apply the cube root to each factor: ∛(27 × 5) = ∛(27) × ∛(5) Simplify ∛(27) = 3 The simplified form is 3 × ∛(5) = 3∛5 The answer is 3∛5.
   Full step-by-step solution

   Step 1: Factor 135 into prime factors: 135 = 27 × 5 Step 2: Since 27 is a perfect cube (3³ = 27), we can write: ∛(135) = ∛(27 × 5) Step 3: Apply the cube root to each factor: ∛(27 × 5) = ∛(27) × ∛(5) Step 4: Simplify ∛(27) = 3 Step 5: The simplified form is 3 × ∛(5) = 3∛5 The answer is 3∛5.

9. ∛(250) + 4∛(2) = ? Answer: 9∛2 Solution: Simplify ∛(250). Factor 250 into perfect cubes: 250 = 125 × 2, and 125 is 5^3. ∛(250) = ∛(125 × 2) = ∛(125) × ∛(2) = 5∛2.
   Full step-by-step solution

   Step 1: Simplify ∛(250). Factor 250 into perfect cubes: 250 = 125 × 2, and 125 is 5^3. Step 2: ∛(250) = ∛(125 × 2) = ∛(125) × ∛(2) = 5∛2. Step 3: The expression becomes 5∛2 + 4∛2. Step 4: Combine like terms: 5∛2 + 4∛2 = (5 + 4)∛2 = 9∛2. The answer is 9∛2.