Root Equations

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Answer Key & Explanations

Root Equations · Grade 8 · Worksheet 3

1. ∛(x - 9) = 4 Answer: 73 Solution: The equation is ∛(x - 9) = 4. To remove the cube root, cube both sides: (∛(x - 9))^3 = 4^3. This simplifies to x - 9 = 64.
   Full step-by-step solution

   Step 1: The equation is ∛(x - 9) = 4. Step 2: To remove the cube root, cube both sides: (∛(x - 9))^3 = 4^3. Step 3: This simplifies to x - 9 = 64. Step 4: Add 9 to both sides: x - 9 + 9 = 64 + 9. Step 5: x = 73. Step 6: Check: ∛(73 - 9) = ∛64 = 4. The answer is 73.

2. Liam is designing a square garden with an area of 256 square feet. He wants to build a path around it that increases the total area to 400 square feet. What will be the side length of the new garden including the path? Answer: 20 feet Solution: Find the side length of the original square garden. The original area is 256 square feet. For a square, area = side × side = side².
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Find the side length of the original square garden.** The original area is 256 square feet. For a square, area = side × side = side². So: side² = 256 side = √256 side = 16 feet. --- **Step 2: Understand the new total area including the path.** The problem says the path around the garden increases the total area to 400 square feet. This means the garden plus the path together form a larger square whose area is 400 square feet. --- **Step 3: Find the side length of the new larger square.** Let the side length of the new larger square (garden + path) be \( x \) feet. Then: x² = 400 x = √400 x = 20 feet. --- **Step 4: Conclusion.** The side length of the new garden including the path is 20 feet. --- **Final answer:** 20 feet

3. Hana is designing a square garden for her community project. The garden has an area of 324 square meters. She also wants to build a cube-shaped compost bin that can hold exactly 216 cubic meters of material. What is the side length of Hana's square garden, and what is the edge length of her cube-shaped compost bin? Answer: 18 and 6 Solution: Find the side length of the square garden. Area of a square = side length squared. 324 = side length squared.
   Full step-by-step solution

   Step 1: Find the side length of the square garden. Area of a square = side length squared. 324 = side length squared. We need to find the square root of 324. 18 x 18 = 324, so the side length is 18 meters. Step 2: Find the edge length of the cube-shaped compost bin. Volume of a cube = edge length cubed. 216 = edge length cubed. We need to find the cube root of 216. 6 x 6 x 6 = 216, so the edge length is 6 meters. Final answer: The side length of the square garden is 18 meters, and the edge length of the cube-shaped compost bin is 6 meters.

4. Noah is building a cube-shaped storage unit for his bedroom. The volume of the cube is 216 cubic inches. He also wants to build a square bulletin board that has an area of 81 square inches to hang above the storage unit. What are the edge length of the cube and the side length of the square bulletin board? Answer: 6 and 9 Solution: Find the edge length of the cube. Volume of a cube = edge length × edge length × edge length = edge length³. So 216 = edge length³.
   Full step-by-step solution

   Step 1: Find the edge length of the cube. Volume of a cube = edge length × edge length × edge length = edge length³. So 216 = edge length³. Take the cube root of both sides: edge length = ∛216. Since 6 × 6 × 6 = 36 × 6 = 216, the cube root of 216 is 6. So the edge length is 6 inches. Step 2: Find the side length of the square bulletin board. Area of a square = side length × side length = side length². So 81 = side length². Take the square root of both sides: side length = √81. Since 9 × 9 = 81, the square root of 81 is 9. So the side length is 9 inches. The cube has edge length 6 inches and the square has side length 9 inches.

5. ∛(125) + √(49) × 2 = ? Answer: 19 Solution: Evaluate the cube root: ∛(125) = 5 Evaluate the square root: √(49) = 7 Perform multiplication: 7 × 2 = 14 Add the results: 5 + 14 = 19 The answer is 19.
   Full step-by-step solution

   Step 1: Evaluate the cube root: ∛(125) = 5 Step 2: Evaluate the square root: √(49) = 7 Step 3: Perform multiplication: 7 × 2 = 14 Step 4: Add the results: 5 + 14 = 19 The answer is 19.

6. Maya is designing a cubic aquarium that needs to hold exactly 729 cubic inches of water. She also wants to create a square mosaic tile for the bottom of the aquarium. If the area of the square tile equals the volume of the aquarium, what is the side length of the cubic aquarium and what is the side length of the square mosaic tile? Answer: 9 Solution: Find the side length of the cubic aquarium. The volume is 729 cubic inches. We need to find the cube root of 729.
   Full step-by-step solution

   Step 1: Find the side length of the cubic aquarium. The volume is 729 cubic inches. We need to find the cube root of 729. Step 2: Calculate cube root of 729: 9 × 9 × 9 = 729, so the side length of the aquarium is 9 inches. Step 3: Find the side length of the square mosaic tile. The area equals the volume of the aquarium, which is 729 square inches. Step 4: Calculate square root of 729: √729 = 27, since 27 × 27 = 729. Step 5: The side length of the cubic aquarium is 9 inches and the side length of the square mosaic tile is 27 inches. The answer is 9 for the aquarium side length and 27 for the tile side length.