Root Equations

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Root Equations · Grade 8 · Worksheet 1

1. Liam is designing a square garden with an area of 289 square feet. He needs to build a fence around the perimeter and also calculate the volume of a cubic storage container that can hold exactly 512 cubic feet of soil. What is the total length of fencing needed for the garden's perimeter, and what is the length of one edge of the storage container? Answer: 68 feet and 8 feet Solution: Let’s go step-by-step. Find the side length of the square garden. We know the area of the square garden is 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 know the area of the square garden is 289 square feet. For a square, Area = side × side = side². So: side² = 289 side = √289 √289 = 17 (since 17 × 17 = 289). So the side length of the garden is 17 feet. --- **Step 2: Find the perimeter of the garden.** The perimeter of a square is: Perimeter = 4 × side. So: Perimeter = 4 × 17 = 68 feet. That means the total length of fencing needed is 68 feet. --- **Step 3: Find the edge length of the cubic storage container.** We are told the container has a volume of 512 cubic feet. For a cube, Volume = edge × edge × edge = edge³. So: edge³ = 512 edge = ∛512. We know 8 × 8 × 8 = 512, so ∛512 = 8. Thus the edge length of the cube is 8 feet. --- **Final Answer:** Fencing length = 68 feet Cube edge length = 8 feet

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (12,5). A square is constructed such that one of its sides lies along the hypotenuse of the triangle, and the square is positioned entirely outside the triangle. What is the area of this square? Answer: 169 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 12 units (horizontal) and 5 units (vertical). Hypotenuse = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 units.
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 12 units (horizontal) and 5 units (vertical). Hypotenuse = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 units. Step 2: The problem states that the square has one side along the hypotenuse, meaning the hypotenuse forms one complete side of the square. Therefore, the side length of the square equals the length of the hypotenuse, which is 13 units. Step 3: Calculate the area of the square. Area of square = side × side = 13 × 13 = 169 square units. The answer is 169.

3. The volume of a cube is 216 cubic centimeters. If each side length of this cube is increased by 2 centimeters, what is the new volume of the cube in cubic centimeters? Answer: 512 Solution: Find the original side length of the cube. The volume of a cube is given by side × side × side = side^3. We know the volume is 216 cm³.
   Full step-by-step solution

   Step 1: Find the original side length of the cube. The volume of a cube is given by side × side × side = side^3. We know the volume is 216 cm³. So, side^3 = 216. We find the cube root of 216: 6 × 6 × 6 = 216, so side = 6 cm. Step 2: Increase the side length by 2 cm. New side length = 6 + 2 = 8 cm. Step 3: Calculate the new volume. New volume = (new side)^3 = 8 × 8 × 8. 8 × 8 = 64, 64 × 8 = 512. Step 4: State the final answer. The new volume is 512 cubic centimeters. Final answer: 512

4. ∛(x - 9) = 4. Solve. Answer: 73 Solution: Start with the equation ∛(x - 9) = 4. Cube both sides to eliminate the cube root: (∛(x - 9))³ = 4³. This simplifies to x - 9 = 64.
   Full step-by-step solution

   Step 1: Start with the equation ∛(x - 9) = 4. Step 2: Cube both sides to eliminate the cube root: (∛(x - 9))³ = 4³. Step 3: This simplifies to x - 9 = 64. Step 4: Add 9 to both sides: x - 9 + 9 = 64 + 9. Step 5: x = 73. The answer is 73.

5. Aisha is designing a square mosaic for her art project. The mosaic has an area of 196 square inches. She also needs to create a cube-shaped display stand that can hold exactly 729 cubic inches of decorative stones. What is the length of one side of the mosaic, and what is the length of one edge of the display stand? 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. Therefore, the mosaic side length is 14 inches. Step 2: Find the edge length of the cube display stand. 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. Therefore, the display stand edge length is 9 inches. Final Answer: The mosaic side length is 14 inches and the display stand edge length is 9 inches.

6. ∛(x + 10) = 5 Answer: 115 Solution: The equation is ∛(x + 10) = 5. To remove the cube root, cube both sides of the equation: (∛(x + 10))³ = 5³. This simplifies to x + 10 = 125.
   Full step-by-step solution

   Step 1: The equation is ∛(x + 10) = 5. Step 2: To remove the cube root, cube both sides of the equation: (∛(x + 10))³ = 5³. Step 3: This simplifies to x + 10 = 125. Step 4: Subtract 10 from both sides: x + 10 - 10 = 125 - 10. Step 5: This gives x = 115. Step 6: Check: ∛(115 + 10) = ∛125 = 5. The answer is 115.

7. A rectangular prism has a length of 12 cm, a width that is the cube root of 64 cm, and a height that is the square root of 144 cm. What is the volume of this prism in cubic centimeters? Answer: 576 Solution: Width = cube root of 64 cube root of 64 = 4 cm (since 4 × 4 × 4 = 64) Height = square root of 144 square root of 144 = 12 cm (since 12 × 12 = 144) Volume = length × width × height Volume = 12 cm × 4 cm × 12 cm Volume = 12 × 4 = 48 48 × 12 = 576 The volume of the prism is 576 cubic centimeters.
   Full step-by-step solution

   Step 1: Find the width of the prism Width = cube root of 64 cube root of 64 = 4 cm (since 4 × 4 × 4 = 64) Step 2: Find the height of the prism Height = square root of 144 square root of 144 = 12 cm (since 12 × 12 = 144) Step 3: Calculate the volume Volume = length × width × height Volume = 12 cm × 4 cm × 12 cm Volume = 12 × 4 = 48 48 × 12 = 576 The volume of the prism is 576 cubic centimeters.