Evaluate Roots

lessonbunny.com

Answer Key & Explanations

Evaluate Roots · Grade 8 · Worksheet 1

1. Emma is designing a community garden with a square plot that has an area of 256 square meters. She wants to build a cubic storage shed for gardening tools where the volume is exactly 1,728 cubic meters. What is the side length of the square garden, and what is the edge length of the cubic storage shed? Answer: 16 and 12 Solution: Find the side length of the square garden. The area is 256 square meters. Since area = side × side, we need to find the square root of 256.
   Full step-by-step solution

   Step 1: Find the side length of the square garden. The area is 256 square meters. Since area = side × side, we need to find the square root of 256. sqrt(256) = 16. So the garden side length is 16 meters. Step 2: Find the edge length of the cubic storage shed. The volume is 1,728 cubic meters. Since volume = edge × edge × edge, we need to find the cube root of 1,728. The cube root of 1,728 is 12 because 12 × 12 × 12 = 1,728. So the shed edge length is 12 meters. Step 3: The side length of the garden is 16 meters and the edge length of the shed is 12 meters.

2. Liam is designing a community garden with a square vegetable patch that has an area of 289 square feet. He wants to build a cubic storage bin for tools 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 cubic storage bin? Answer: 17 feet and 8 feet Solution: Find the side length of the square vegetable patch. We are told the area is 289 square feet. Area = side × side = side².
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the side length of the square vegetable patch.** We are told the area is 289 square feet. For a square, Area = side × side = side². So: side² = 289 side = √289. We know 17 × 17 = 289, so: side = 17 feet. --- **Step 2: Find the edge length of the cubic storage bin.** We are told the volume is 512 cubic feet. For a cube, Volume = edge × edge × edge = edge³. So: edge³ = 512. We need the cube root of 512. Let's check: 8 × 8 = 64, 64 × 8 = 512. So edge = 8 feet. --- **Step 3: Final answer.** Square garden side length = 17 feet. Cube storage bin edge length = 8 feet. --- **Answer:** 17 feet and 8 feet

3. ∛(125) + √(64) = ? Answer: 13 Solution: We need to calculate the cube root of 125 and the square root of 64, then add them together. Calculate the cube root of 125.
   Full step-by-step solution

   Step 1: Understand the problem. We need to calculate the cube root of 125 and the square root of 64, then add them together. Step 2: Calculate the cube root of 125. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. We ask: what number multiplied by itself three times equals 125? 5 * 5 = 25, and 25 * 5 = 125. So, cube root of 125 = 5. Step 3: Calculate the square root of 64. The square root of a number is a value that, when multiplied by itself, gives the original number. We ask: what number multiplied by itself equals 64? 8 * 8 = 64. So, square root of 64 = 8. Step 4: Add the two results. 5 + 8 = 13. Step 5: Final answer. The result is 13.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). What is the length of the hypotenuse of this triangle? Answer: 10 Solution: Identify the coordinates of the triangle's vertices: A(0,0), B(6,0), C(6,8) Calculate the length of side AB: Since both points have y=0, the length is the difference in x-coordinates: 6 - 0 = 6 Calculate the length of side BC: Since both points have x=6, the length is the difference in…
   Full step-by-step solution

   Step 1: Identify the coordinates of the triangle's vertices: A(0,0), B(6,0), C(6,8) Step 2: Calculate the length of side AB: Since both points have y=0, the length is the difference in x-coordinates: 6 - 0 = 6 Step 3: Calculate the length of side BC: Since both points have x=6, the length is the difference in y-coordinates: 8 - 0 = 8 Step 4: Apply the Pythagorean theorem: hypotenuse² = 6² + 8² = 36 + 64 = 100 Step 5: Find the square root: hypotenuse = sqrt(100) = 10 The answer is 10.

5. Aisha is designing a community garden with a square plot that has an area of 324 square meters. She wants to build a cubic water tank that has the same side length as the garden plot. What is the volume, in cubic meters, of the water tank Aisha is planning to build? Answer: 5832 Solution: Find the side length of the square garden. Since area = side × side, we need to find what number squared equals 324. The square root of 324 is 18, because 18 × 18 = 324.
   Full step-by-step solution

   Step 1: Find the side length of the square garden. Since area = side × side, we need to find what number squared equals 324. Step 2: The square root of 324 is 18, because 18 × 18 = 324. So the side length is 18 meters. Step 3: Calculate the volume of the cubic water tank. Volume of a cube = side × side × side. Step 4: Volume = 18 × 18 × 18 = 324 × 18 = 5832. Step 5: The volume of the water tank is 5832 cubic meters.

6. Emma is designing a community garden with a square plot that has an area of 324 square meters. She wants to build a cubic compost bin where the volume is exactly 1,728 cubic meters. What is the side length of the square garden, and what is the edge length of the cubic compost bin? Answer: 18 and 12 Solution: Find the side length of the square garden. The area is 324 square meters, so side length = sqrt(324). Since 18 × 18 = 324, the side length is 18 meters.
   Full step-by-step solution

   Step 1: Find the side length of the square garden. The area is 324 square meters, so side length = sqrt(324). Since 18 × 18 = 324, the side length is 18 meters. Step 2: Find the edge length of the cubic compost bin. The volume is 1,728 cubic meters, so edge length = cube root of 1,728. Since 12 × 12 × 12 = 1,728, the edge length is 12 meters. Step 3: The side length of the garden is 18 meters and the edge length of the compost bin is 12 meters.

7. Liam is designing a cubic container to hold exactly 512 cubic centimeters of liquid. He needs to know the length of each side of the cube to order the correct materials. What is the side length, in centimeters, of Liam's container? Answer: 8 Solution: We know the container is a cube and holds 512 cubic centimeters of liquid. That means the volume of the cube is 512 cm³. Write the formula for the volume of a cube.
   Full step-by-step solution

   We know the container is a cube and holds 512 cubic centimeters of liquid. That means the volume of the cube is 512 cm³. Step 1: Write the formula for the volume of a cube. Volume = side length × side length × side length If we call the side length \( s \), then: Volume = s³ Step 2: Substitute the given volume into the formula. s³ = 512 Step 3: Find the cube root of 512 to solve for s. We want the number that, when multiplied by itself three times, gives 512. Let’s test some whole numbers: - 5³ = 125 (too small) - 6³ = 216 (too small) - 7³ = 343 (too small) - 8³ = 8 × 8 × 8 = 64 × 8 = 512 (correct) Step 4: Conclusion The side length s = 8 cm. Final answer: 8

8. Liam is designing a cubic storage container that needs to hold exactly 64 cubic meters of water. He wants to calculate the exact side length of this cube. What is the side length in meters? Answer: 4 Solution: We are told the container is a cube and holds 64 cubic meters of water. That means the volume of the cube is 64 m³. Recall the formula for the volume of a cube.
   Full step-by-step solution

   We are told the container is a cube and holds 64 cubic meters of water. That means the volume of the cube is 64 m³. Step 1: Recall the formula for the volume of a cube. Volume = side length × side length × side length If we call the side length \( s \), then: Volume = s³ Step 2: Substitute the given volume into the formula. s³ = 64 Step 3: Find the side length \( s \) by taking the cube root of both sides. s = cube root of 64 Step 4: Determine what number multiplied by itself three times gives 64. Check possible whole numbers: 2³ = 8 (too small) 3³ = 27 (too small) 4³ = 4 × 4 × 4 = 16 × 4 = 64 (correct) Step 5: Conclusion. The side length is 4 meters. Final answer: 4