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. Mason is building a cube-shaped water tank for his school's science fair. The tank needs to hold exactly 512 cubic inches of water. He also wants to create a square observation window on one side of the tank with an area of 81 square inches. What is the edge length of the cube tank, and what is the side length of the square window? Answer: 8 and 9 Solution: Find the edge length of the cube tank. Volume of a cube = edge length cubed. So edge length = cube root of 512.
   Full step-by-step solution

   Step 1: Find the edge length of the cube tank. Volume of a cube = edge length cubed. So edge length = cube root of 512. Since 8 x 8 x 8 = 512, the edge length is 8 inches. Step 2: Find the side length of the square window. Area of a square = side length squared. So side length = square root of 81. Since 9 x 9 = 81, the side length is 9 inches. The cube tank has edge length 8 inches and the square window has side length 9 inches.

3. ∛(125) × √(49) + 3² = ? Answer: 44 Solution: Calculate the cube root of 125: ∛(125) = 5 Calculate the square root of 49: √(49) = 7 Multiply the results: 5 × 7 = 35 Calculate 3 squared: 3² = 9 Add the results: 35 + 9 = 44 The final answer is 44.
   Full step-by-step solution

   Step 1: Calculate the cube root of 125: ∛(125) = 5 Step 2: Calculate the square root of 49: √(49) = 7 Step 3: Multiply the results: 5 × 7 = 35 Step 4: Calculate 3 squared: 3² = 9 Step 5: Add the results: 35 + 9 = 44 Step 6: The final answer is 44.

4. ∛(125) + √(49) - 2² = ? Answer: 8 Solution: Calculate the cube root of 125: ∛(125) = 5 because 5 × 5 × 5 = 125 Calculate the square root of 49: √(49) = 7 because 7 × 7 = 49 Calculate 2 squared: 2² = 4 Substitute the values: 5 + 7 - 4 Add first: 5 + 7 = 12 Subtract: 12 - 4 = 8 The answer is 8.
   Full step-by-step solution

   Step 1: Calculate the cube root of 125: ∛(125) = 5 because 5 × 5 × 5 = 125 Step 2: Calculate the square root of 49: √(49) = 7 because 7 × 7 = 49 Step 3: Calculate 2 squared: 2² = 4 Step 4: Substitute the values: 5 + 7 - 4 Step 5: Add first: 5 + 7 = 12 Step 6: Subtract: 12 - 4 = 8 The answer is 8.

5. ∛(x + 7) = 9. Solve for x. Answer: 722 Solution: The equation is ∛(x + 7) = 9. To eliminate the cube root, cube both sides of the equation: (∛(x + 7))³ = 9³. This simplifies to x + 7 = 729.
   Full step-by-step solution

   Step 1: The equation is ∛(x + 7) = 9. Step 2: To eliminate the cube root, cube both sides of the equation: (∛(x + 7))³ = 9³. Step 3: This simplifies to x + 7 = 729. Step 4: Subtract 7 from both sides: x + 7 - 7 = 729 - 7. Step 5: x = 722. The answer is 722.

6. Liam is designing a square garden with an area of 289 square feet. He wants to build a stone path around the perimeter, but needs to know the side length first. What is the length of one side of Liam's garden? Answer: 17 Solution: We are told the garden is square and has an area of 289 square feet. Area = side × side = side^2 Let s be the side length in feet.
   Full step-by-step solution

   We are told the garden is square and has an area of 289 square feet. Step 1: Recall that for a square, the area is given by: Area = side × side = side^2 Step 2: Let s be the side length in feet. Then: s^2 = 289 Step 3: To find s, we take the square root of both sides: s = sqrt(289) Step 4: We need to find the number that, when multiplied by itself, gives 289. Check possible squares: 10 × 10 = 100 (too small) 15 × 15 = 225 (too small) 17 × 17 = 289 (yes, that works) 20 × 20 = 400 (too large) Step 5: So, s = 17. Step 6: Check: 17 × 17 = 289, which matches the given area. Therefore, the length of one side of Liam's garden is 17 feet.