Root Equations

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Root Equations · Grade 8 · Worksheet 2

1. Liam is designing a square garden with an area of 289 square feet. He wants to build a cube-shaped planter box where the volume is exactly 512 cubic feet. What is the side length of the square garden, and what is the edge length of the cube planter? Answer: 17 feet and 8 feet Solution: Let’s go step by step. Find the side length of the square garden. We are told the garden is a square with area 289 square feet.
   Full step-by-step solution

   Let’s go step by step. **Step 1: Find the side length of the square garden.** We are told the garden is a square with area 289 square feet. For a square: Area = side × side = side² So: side² = 289 To find the side length, take the square root of 289: side = √289 We know 17 × 17 = 289, so: side = 17 feet. **Step 2: Find the edge length of the cube planter.** We are told the planter is a cube with volume 512 cubic feet. For a cube: Volume = edge × edge × edge = edge³ So: edge³ = 512 To find the edge length, take the cube root of 512. We can test possible numbers: 8 × 8 = 64, and 64 × 8 = 512. So: edge = 8 feet. **Final Answer:** Square garden side length = 17 feet Cube planter edge length = 8 feet

2. Maya is designing a square mosaic for her art project. The mosaic has an area of 196 square inches. She also needs to calculate the side length of a cube-shaped storage box that can hold exactly 729 cubic inches of art supplies. What is the side length of Maya's square mosaic, and what is the edge length of the cube storage box? Answer: 14 and 9 Solution: Find the side length of the square mosaic. The area of a square is side × side = side². We know the area is 196 square inches.
   Full step-by-step solution

   Step 1: Find the side length of the square mosaic. The area of a square is side × side = side². We know the area is 196 square inches. So, side² = 196. The square root of 196 is 14, because 14 × 14 = 196. The side length of the mosaic is 14 inches. Step 2: Find the edge length of the cube storage box. The volume of a cube is edge × edge × edge = edge³. We know the volume is 729 cubic inches. So, edge³ = 729. The cube root of 729 is 9, because 9 × 9 × 9 = 729. The edge length of the storage box is 9 inches. Step 3: State the final answers. The side length of the mosaic is 14 inches, and the edge length of the cube is 9 inches.

3. √(144) + ∛(125) - 2² = ? Answer: 13 Solution: Calculate the square root: √(144) = 12 Calculate the cube root: ∛(125) = 5 Calculate the exponent: 2² = 4 Substitute back into the expression: 12 + 5 - 4 Perform addition first: 12 + 5 = 17 Perform subtraction: 17 - 4 = 13 The answer is 13.
   Full step-by-step solution

   Step 1: Calculate the square root: √(144) = 12 Step 2: Calculate the cube root: ∛(125) = 5 Step 3: Calculate the exponent: 2² = 4 Step 4: Substitute back into the expression: 12 + 5 - 4 Step 5: Perform addition first: 12 + 5 = 17 Step 6: Perform subtraction: 17 - 4 = 13 The answer is 13.

4. Emma is designing a cubic aquarium for her science class that needs to hold exactly 216 liters of water. She also needs to calculate the side length of a square display area that has the same area as one face of the aquarium. What is the side length of the cubic aquarium, and what is the side length of the square display area? Answer: 6 Solution: Volume of cube = side × side × side = side³ 216 = side³ side = cube root of 216 Since 6 × 6 × 6 = 216, the side length is 6 cm Area of square face = side × side = 6 × 6 = 36 cm² Area of square = side × side = 36 side = square root of 36 Since 6 × 6 = 36, the side length is 6 cm The side length…
   Full step-by-step solution

   Step 1: Find the side length of the cubic aquarium Volume of cube = side × side × side = side³ 216 = side³ side = cube root of 216 Since 6 × 6 × 6 = 216, the side length is 6 cm Step 2: Find the area of one face of the aquarium Area of square face = side × side = 6 × 6 = 36 cm² Step 3: Find the side length of the square display area Area of square = side × side = 36 side = square root of 36 Since 6 × 6 = 36, the side length is 6 cm The side length of both the cubic aquarium and the square display area is 6 cm.

5. Aisha is designing a square mosaic for her art project. The mosaic has an area of 196 square inches. She also needs to build a cubic display case that can hold exactly 729 cubic inches. What is the side length of the square mosaic, and what is the edge length of the cubic display case? Answer: 14 and 9 Solution: Find the side length of the square mosaic. The area of a square is side length squared, so side length = sqrt(area). Area = 196 square inches, so side length = sqrt(196) = 14 inches.
   Full step-by-step solution

   Step 1: Find the side length of the square mosaic. The area of a square is side length squared, so side length = sqrt(area). Area = 196 square inches, so side length = sqrt(196) = 14 inches. Step 2: Find the edge length of the cubic display case. The volume of a cube is edge length cubed, so edge length = cube root of volume. Volume = 729 cubic inches, so edge length = cube root of 729. Since 9 × 9 × 9 = 729, the edge length = 9 inches. Step 3: State both answers. The side length of the mosaic is 14 inches, and the edge length of the display case is 9 inches.