Radicals & Exponents

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Answer Key & Explanations

Radicals & Exponents · Grade 9 · Worksheet 3

1. √32 × √8 = ? Answer: 16 Solution: Multiply the numbers under the radicals: √32 × √8 = √(32 × 8) Calculate 32 × 8 = 256 Simplify √256 Since 256 is a perfect square (16 × 16 = 256), √256 = 16 The answer is 16.
   Full step-by-step solution

   Step 1: Multiply the numbers under the radicals: √32 × √8 = √(32 × 8) Step 2: Calculate 32 × 8 = 256 Step 3: Simplify √256 Step 4: Since 256 is a perfect square (16 × 16 = 256), √256 = 16 The answer is 16.

2. √(96) + √(6) = ? Answer: 5√6 Solution: Simplify √(96) Factor 96: 96 = 16 × 6 √(96) = √(16 × 6) = √16 × √6 = 4√6 √(96) + √(6) = 4√6 + √6 Both terms have √6, so we add the coefficients: 4√6 + 1√6 = (4 + 1)√6 = 5√6 The answer is 5√6.
   Full step-by-step solution

   Step 1: Simplify √(96) Factor 96: 96 = 16 × 6 √(96) = √(16 × 6) = √16 × √6 = 4√6 Step 2: Write the expression with simplified terms √(96) + √(6) = 4√6 + √6 Step 3: Combine like terms Both terms have √6, so we add the coefficients: 4√6 + 1√6 = (4 + 1)√6 = 5√6 The answer is 5√6.

3. Kaia is building a rectangular planter box for her garden. The length of the box is 4√12 feet and the width is 3√27 feet. She needs to calculate the area of the base to determine how much soil to buy. What is the area of the base in simplest radical form? Answer: B. 216 Solution: Write the area formula: Area = length × width = (4√12) × (3√27) Multiply the coefficients: 4 × 3 = 12 Multiply the radicals: √12 × √27 = √(12 × 27) = √324 Simplify √324: Since 324 = 18², √324 = 18 Multiply results: 12 × 18 = 216 The area is 216 square feet The correct answer is 216.
   Full step-by-step solution

   Step 1: Write the area formula: Area = length × width = (4√12) × (3√27) Step 2: Multiply the coefficients: 4 × 3 = 12 Step 3: Multiply the radicals: √12 × √27 = √(12 × 27) = √324 Step 4: Simplify √324: Since 324 = 18², √324 = 18 Step 5: Multiply results: 12 × 18 = 216 Step 6: The area is 216 square feet The correct answer is 216.

4. Sophia is creating a square mosaic with an area of 96 square inches. She wants to know the exact length of each side in simplest radical form. What is the side length of Sophia's mosaic? Answer: 4√6 Solution: The area of a square is side length squared, so side = √(area) Side = √96 Factor 96 into perfect squares: 96 = 16 × 6 √96 = √(16 × 6) = √16 × √6 √16 = 4, so side = 4√6 The exact side length in simplest radical form is 4√6 inches
   Full step-by-step solution

   Step 1: The area of a square is side length squared, so side = √(area) Step 2: Side = √96 Step 3: Factor 96 into perfect squares: 96 = 16 × 6 Step 4: √96 = √(16 × 6) = √16 × √6 Step 5: √16 = 4, so side = 4√6 Step 6: The exact side length in simplest radical form is 4√6 inches

5. ∛(54) + ∛(16) = ? Answer: 5∛2 Solution: Simplify ∛(54) 54 = 27 × 2 = 3³ × 2 ∛(54) = ∛(27 × 2) = ∛(27) × ∛(2) = 3∛2 Simplify ∛(16) 16 = 8 × 2 = 2³ × 2 ∛(16) = ∛(8 × 2) = ∛(8) × ∛(2) = 2∛2 3∛2 + 2∛2 = (3 + 2)∛2 = 5∛2 The answer is 5∛2.
   Full step-by-step solution

   Step 1: Simplify ∛(54) 54 = 27 × 2 = 3³ × 2 ∛(54) = ∛(27 × 2) = ∛(27) × ∛(2) = 3∛2 Step 2: Simplify ∛(16) 16 = 8 × 2 = 2³ × 2 ∛(16) = ∛(8 × 2) = ∛(8) × ∛(2) = 2∛2 Step 3: Add the simplified terms 3∛2 + 2∛2 = (3 + 2)∛2 = 5∛2 The answer is 5∛2.

6. Mason is building a square garden with an area of 98 square feet. He needs to know the exact length of one side of the garden to buy the right amount of fencing. What is the simplified radical expression for the side length of Mason's garden? Answer: 7*sqrt(2) Solution: The area of a square is side length squared, so side = sqrt(area) Mason's garden has area 98, so side = sqrt(98) Factor 98 into 49 * 2, where 49 is a perfect square sqrt(98) = sqrt(49 * 2) = sqrt(49) * sqrt(2) sqrt(49) = 7 Therefore, the simplified radical is 7*sqrt(2) The exact side length is…
   Full step-by-step solution

   Step 1: The area of a square is side length squared, so side = sqrt(area) Step 2: Mason's garden has area 98, so side = sqrt(98) Step 3: Factor 98 into 49 * 2, where 49 is a perfect square Step 4: sqrt(98) = sqrt(49 * 2) = sqrt(49) * sqrt(2) Step 5: sqrt(49) = 7 Step 6: Therefore, the simplified radical is 7*sqrt(2) The exact side length is 7*sqrt(2) feet.

7. Emma is building a square tabletop with an area of 75 square inches. She needs to cut wooden trim pieces to frame the tabletop. Each side length can be expressed as √(75) inches. What is the simplified radical form of the side length? Answer: 5√3 Solution: Start with √(75) Factor 75 into prime factors: 75 = 25 × 3 Recognize that 25 is a perfect square: √(75) = √(25 × 3) Apply the property √(a×b) = √a × √b: √(25) × √(3) Simplify √(25) to 5: 5 × √(3) Write the simplified form: 5√3 The answer is 5√3.
   Full step-by-step solution

   Step 1: Start with √(75) Step 2: Factor 75 into prime factors: 75 = 25 × 3 Step 3: Recognize that 25 is a perfect square: √(75) = √(25 × 3) Step 4: Apply the property √(a×b) = √a × √b: √(25) × √(3) Step 5: Simplify √(25) to 5: 5 × √(3) Step 6: Write the simplified form: 5√3 The answer is 5√3.

8. √(48) × √(12) = ? Answer: 24 Solution: Multiply the numbers under the square roots: √(48) × √(12) = √(48 × 12) Calculate 48 × 12 = 576 Simplify √576 = 24 The final answer is 24.
   Full step-by-step solution

   Step 1: Multiply the numbers under the square roots: √(48) × √(12) = √(48 × 12) Step 2: Calculate 48 × 12 = 576 Step 3: Simplify √576 = 24 Step 4: The final answer is 24.