Download Grade 8 Cubes and Cube Root

ID : my-8-Cubes-and-Cube-Root [1]
Grade 8
Cubes and Cube Root
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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
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-2
7
.
b.
17
c.
-7
17
18
d.
289
18
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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
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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
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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.
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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
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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.
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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
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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
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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.
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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
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