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ID : my-8-Cubes-and-Cube-Root [1] Grade 8 Cubes and Cube Root For more such worksheets visit www.edugain.com Answer t he quest ions (1) Calculate the cube root of 729 (2) Solve the f ollowing: (3) What is the value of 123? (4) If you subtract a number x f rom 37 times that number, and then take the cube of the dif f erence, what will the result be? (5) T ake a number x, and multiply it with 5. T ake the cube of the resulting number. What is the ratio of this number to the cube of the original number? (6) If the volume of a cube is 729 m3, then what is the length of each side of the cube? (7) Which of the f ollowing statements is f alse: 1000 is not a perfect cube. Cubes of odd numbers are always odd. 98 is not a perfect cube. 216 is a perfect cube. (8) If you have a container in the shape of a cube that has a volume of 64000 m3, then what is the area of each of the f aces of the cube? (9) A cubical box has all sides of 20 m. What is its volume? Choose correct answer(s) f rom given choice (10) What is the value of a. . -3 b. 2 c. 3 -2 d. 3 (11) What is the value of a. 18 17 16 (C) 2016 Edugain (www.Edugain.com) -2 7 . b. 17 c. -7 17 18 d. 289 18 Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [2] (12) Which of the f ollowing numbers is a perf ect cube? a. 2746 b. 2197 c. 39310 d. 4911 (13) If you subtract a number x f rom 28 times that number, and then take the cube of the dif f erence, what will be the result? a. 784x3 b. 19683x3 c. 25x3 d. 794x2 Fill in t he blanks (14) (15) If a cube has a surf ace area of 4374 m2, then the volume of the cube is m3 T he cube root of 37 correct to three places of decimal is © 2016 Edugain (www.edugain.com). All Rights Reserved (C) 2016 Edugain (www.Edugain.com) Many more such worksheets can be generated at www.edugain.com Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [3] Answers (1) 9 Step 1 We have been asked to f ind the cube root of 729. Step 2 Prime f actors of 729 are, 3 × 3 × 3 × 3 × 3 × 3. Step 3 Now, make group of three prime f actors. (3 × 3 × 3) × (3 × 3 × 3) Step 4 T here is no prime f actor lef t which is not part of any group. Cube root is given by the multiplication of a element f rom each group of prime f actors. i.e. 3×3 =9 Step 5 T heref ore, the cube root of 729 is 9. (2) -90 Step 1 Let's f irst f ind prime f actors of -1350, -1350 = -1 × 2 × 3 × 3 × 3 × 5 × 5 Step 2 Similarly f ind prime f actors of 540, 540 = 2 × 2 × 3 × 3 × 3 × 5 Step 3 Now, ³√(-1350) ³√(540) = ³√(-1 × 2 × 3 × 3 × 3 × 5 × 5) ³√(2 × 2 × 3 × 3 × 3 × 5) = ³√(-1 × 2 × 3 × 3 × 3 × 5 × 5 × 2 × 2 × 3 × 3 × 3 × 5) = ³√[-1 × (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3) × (5 × 5 × 5)] = ³√[-13 × 23 × 33 × 33 × 53] = (-1 × 2 × 3 × 3 × 5) = -90 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [4] (3) 1728 Step 1 According to the question, we have to f ind the value of 123. Step 2 Now, 123 = 12 × 12 × 12 = 144 × 12 = 1728 Step 3 T heref ore, the value of 123 is 1728. (4) 46656x3 Step 1 According to the question, the f irst number is x and the second number is 37 x . Step 2 Dif f erence of two the numbers = (37 x - x) Step 3 Cube of the dif f erence = (37 x - x)3 = (36 x)3 = 46656 x3 Step 4 T heref ore, the result will be 46656x3. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [5] (5) 125:1 Step 1 If we multiply the number x with 5, the resulting number is: 5x. Step 2 T he cube of the resulting number 5x = (5x)3 = 125x3 Step 3 T he original number is x. Cube of the original number = x3. Step 4 T he ratio of the new number to the original number = New number Original number = 125x3 x3 = 125 1 Step 5 T heref ore, the ratio of the new number to the cube of the original number is 125:1. (6) 9 m Step 1 We know that the volume of a cube with side a = a3 T he given volume of the cube = 729 m3 Step 2 Let us write these two f acts as an equation and f ind the value of a: a3 = 729 m3 ⇒ a3 = 93 m3 ⇒a=9m Step 3 T heref ore, the length of each side of the cube is 9 m. (7) 1000 is not a perf ect cube (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [6] (8) 1600 m2 Step 1 Let us assume the side of a cube shape container is a. Step 2 Volume of cube = 64000 m3 also, volume of the cube = a3 compare both, a3 = 64000 ⇒ a = (64000) 1 3 ⇒ a = 40 Step 3 Area of side of the cube = a2 = 402 = 1600 Step 4 T heref ore, the area of the f ace of cube shape container is 1600 m2. (9) 8000 m3 Step 1 We know that the volume of a cube (or a cubical box) is calculated by taking the cube of its side. Step 2 We have been told that the side of the cube in question is 20 m. T heref ore, its volume = (20)3 m3 = 20 × 20 × 20 m3 = 8000 m3 Step 3 T heref ore, the volume of the cubical box is 8000 m3. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [7] (10) -7 b. 3 Step 1 Let's f irst f ind prime f actors of 686, 686 = 7 × 7 × 7 × 2 Step 2 Similarly prime f actors of 54, 54 = 2 × 3 × 3 × 3 Step 3 Now, = = = -7 3 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [8] (11) 17 b. 18 Step 1 We have been asked to f ind the value of . Step 2 Now, =( (17 × 17 × 17) (18 × 18 × 18) (17)3 =( ) (18)3 = (( 17 18 = )3) ) 1 3 1 3 1 3 17 18 Step 3 T heref ore, the value of is 17 . 18 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [9] (12) b. 2197 Step 1 T he def inition of a perf ect cube is a number that is the result of multiplying an integer by itself three times.Steps to f ind the perf ect cube are as f ollows. i. Resolve the given number into prime f actors. ii. Group the f actors in 3 in such a way that each number of the group is same. iii. If there is nothing lef t with the group of prime numbers so the number is perf ect cube otherwise not. Step 2 Prime f actors of 2197 = 13 x 13 x 13 T here is no prime f actor lef t with the group of prime f actors. So, 2197 is a perf ect cube. Step 3 Prime f actors of 39310 = 2 x 5 x 3931 T here is a prime f actor lef t. So, 39310 is not a perf ect cube. Step 4 Prime f actors of 4911 = 3 x 1637 T here is a prime f actor lef t. So, 4911 is not a perf ect cube. Step 5 Prime f actors of 2746 = 2 x 1373 T here is a prime f actor lef t. So, 2746 is not a perf ect cube. Step 6 T heref ore, 2197 is the perf ect cube. (13) b. 19683x3 Step 1 According to the question, the f irst number is x and the second number is 28 x. Step 2 Dif f erence of the numbers = (28 x - x) Step 3 Cube of the dif f erence = (28 x - x)3 = (27 x)3 = 19683 x3 Step 4 T heref ore, the result will be 19683x3. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : my-8-Cubes-and-Cube-Root [10] (14) 19683 Step 1 We need to f ind the volume of the cube. T he volume of a cube with side a is a3. T his means if we can f ind the value of a, we can easily f ind the value of the volume of the cube. Step 2 Let us f ind the value of the side a of the cube. We have been told that the surf ace area of the cube is 4374 m2. Let us use this f act to write an equation and f ind the value of a. Step 3 T he surf ace area of the cube with side a = 6a2. T he given surf ace area of the cube = 4374 m2. T his means, 4374 = 6a2 4374 ⇒ a2 = 6 ⇒ a2 = 729 ⇒ a = √729 ⇒ a = 27 Step 4 Now that we know that the value of the side of the cube is 27, let us f ind the volume of the cube, which is = a3 = (27)3 = 19683 m3 Step 5 T heref ore, the volume of the cube is 19683 m3. (15) 3.332 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited