Root Equations

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Answer Key & Explanations

Root Equations · Grade 8 · Worksheet 3

1. 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 edge 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 mosaic, and what is the edge length of the storage box? Answer: 14 and 9 Solution: Find the side length of the square mosaic. The area is 196 square inches. Since area = side × side, we need to find the square root of 196.
   Full step-by-step solution

   Step 1: Find the side length of the square mosaic. The area is 196 square inches. Since area = side × side, we need to find the square root of 196. sqrt(196) = 14. So the mosaic side length is 14 inches. Step 2: Find the edge length of the cube storage box. The volume is 729 cubic inches. Since volume = edge × edge × edge, we need to find the cube root of 729. The cube root of 729 is 9 because 9 × 9 × 9 = 729. So the storage box edge length is 9 inches. Step 3: The final answer is the mosaic side length (14) and the storage box edge length (9).

2. ∛(x - 10) = 5. Solve for x. Answer: 135 Solution: Start with the equation ∛(x - 10) = 5. Cube both sides to eliminate the cube root: (∛(x - 10))³ = 5³. This simplifies to x - 10 = 125.
   Full step-by-step solution

   Step 1: Start with the equation ∛(x - 10) = 5. Step 2: Cube both sides to eliminate the cube root: (∛(x - 10))³ = 5³. Step 3: This simplifies to x - 10 = 125. Step 4: Add 10 to both sides: x - 10 + 10 = 125 + 10. Step 5: x = 135. Step 6: Check: ∛(135 - 10) = ∛125 = 5. The answer is 135.

3. ∛(x - 7) = 5. Solve for x. Answer: 132 Solution: Start with the equation ∛(x - 7) = 5. Cube both sides to eliminate the cube root: (∛(x - 7))³ = 5³. This simplifies to x - 7 = 125.
   Full step-by-step solution

   Step 1: Start with the equation ∛(x - 7) = 5. Step 2: Cube both sides to eliminate the cube root: (∛(x - 7))³ = 5³. Step 3: This simplifies to x - 7 = 125. Step 4: Add 7 to both sides: x - 7 + 7 = 125 + 7. Step 5: x = 132. The answer is 132.

4. √(64) + ∛(27) = ? Answer: 11 Solution: √(64) + ∛(27) Evaluate √(64) The square root of 64 is the number that, when multiplied by itself, gives 64. 8 × 8 = 64 So √(64) = 8.
   Full step-by-step solution

   Let's solve step by step. Step 1: Understand the problem We need to compute: √(64) + ∛(27) Step 2: Evaluate √(64) The square root of 64 is the number that, when multiplied by itself, gives 64. 8 × 8 = 64 So √(64) = 8. Step 3: Evaluate ∛(27) The cube root of 27 is the number that, when multiplied by itself three times, gives 27. 3 × 3 × 3 = 27 So ∛(27) = 3. Step 4: Add the results 8 + 3 = 11. Step 5: Final answer The correct answer is 11.

5. Aisha is designing a square mosaic for her art project. The mosaic has an area of 196 square inches. She wants to create a cube-shaped display case where the volume equals the area of her mosaic. What should be the side length of the cube-shaped display case? Answer: 6 Solution: The mosaic has an area of 196 square inches, and the cube's volume equals this area, so the cube's volume is 196 cubic inches. For a cube, volume = side length × side length × side length, or V = s³.
   Full step-by-step solution

   Step 1: The mosaic has an area of 196 square inches, and the cube's volume equals this area, so the cube's volume is 196 cubic inches. Step 2: For a cube, volume = side length × side length × side length, or V = s³. Step 3: Set up the equation: s³ = 196 Step 4: To find s, take the cube root of both sides: s = cube root of 196 Step 5: Calculate cube root of 196: 5 × 5 × 5 = 125, 6 × 6 × 6 = 216, so the cube root is between 5 and 6. Step 6: Try 5.8: 5.8 × 5.8 = 33.64, 33.64 × 5.8 = 195.112 Step 7: Try 5.81: 5.81 × 5.81 = 33.7561, 33.7561 × 5.81 = 196.12 (slightly over) Step 8: Try 5.808: 5.808 × 5.808 = 33.732864, 33.732864 × 5.808 = 196.00 Step 9: Since the problem asks for a reasonable measurement, we round to the nearest whole number: 6 inches. The answer is 6.

6. √(144) + ∛(125) = ? Answer: 17 Solution: Find the square root of 144. Since 12 × 12 = 144, √(144) = 12 Find the cube root of 125. Since 5 × 5 × 5 = 125, ∛(125) = 5 Add the results: 12 + 5 = 17 The answer is 17.
   Full step-by-step solution

   Step 1: Find the square root of 144. Since 12 × 12 = 144, √(144) = 12 Step 2: Find the cube root of 125. Since 5 × 5 × 5 = 125, ∛(125) = 5 Step 3: Add the results: 12 + 5 = 17 The answer is 17.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A square is constructed on the hypotenuse of this triangle such that the hypotenuse forms one side of the square. What is the area of this square? Answer: 100 Solution: Identify the triangle's sides. The vertices are (0,0), (6,0), and (6,8). From (0,0) to (6,0): length = 6 (horizontal leg).
   Full step-by-step solution

   Step 1: Identify the triangle's sides. The vertices are (0,0), (6,0), and (6,8). From (0,0) to (6,0): length = 6 (horizontal leg). From (6,0) to (6,8): length = 8 (vertical leg). From (0,0) to (6,8): this is the hypotenuse. Step 2: Calculate the hypotenuse length. Using the Pythagorean theorem: hypotenuse^2 = 6^2 + 8^2 = 36 + 64 = 100. So hypotenuse = sqrt(100) = 10. Step 3: Understand the square construction. The hypotenuse of length 10 is one side of the square. So the square has side length 10. Step 4: Find the area of the square. Area of a square = side^2 = 10^2 = 100. Step 5: Final answer. The area of the square is 100.