Download Grade 8 Cubes and Cube Root

ID : ph-8-Cubes-and-Cube-Root [1]
Grade 8
Cubes and Cube Root
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Answer t he quest ions
(1)
What is the value of
(2)
If you have a container in the shape of a cube that has a volume of 2197 m3, then what is the
area of each of the f aces of the cube?
(3)
T he number of people living in a small town is f ound to be a perf ect cube. We know that the
number of men in the town is 31207, and the number of women in the town is 43329.
By an odd coincidence, we count and f ind that the number of children in the town is also a
perf ect cube. If we give you a hint that this number is more than 10640, then what is the
smallest possible number of children in the town?
(4) What is the value of 37 3?
(5)
T here are three numbers a, b and c.
b is 4 times a.
c is 0.2 times b.
T he sums of their cubes if 524.096.
Find the value of b
(6) Which of the f ollowing numbers is not a perf ect cube?
1728, 432, 2744, 729
(7)
What is the value of (
5
)3
6
(8)
T ake a number x, and multiply it with 6. T ake the cube of the resulting number. What is the ratio
of this number to the cube of the original number?
(9) Find the value of
Choose correct answer(s) f rom given choice
(10) If you add a number x with another number that is 16 times the value of x, and then take the
cube of the sum, what will be the result?
a. 262x2
b. 256x3
c. 4743x3
d. 4913x3
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ID : ph-8-Cubes-and-Cube-Root [2]
(11) Which of the f ollowing statements is f alse:
a. T he sum of the cubes of the f irst n natural
numbers is equal to the square of their sum
b. If a negative integer is a perf ect cube, then
the corresponding positive integer will also be
a perf ect cube
c. For an integer, the cube root will always be
d. A number that has the last digit as 1 will
greater than the square root
have 1 as the last digit of its cube
(12) Solve the f ollowing:
a. -40
b. 37
c. 1600
d. -37
(13) Which of the f ollowing numbers is a perf ect cube?
a. 26999
b. 15629
c. 29792
d. 32768
Fill in t he blanks
(14)
T here are two numbers such that sum of the numbers is 28 and their dif f erence is 4.
is
the sum of their cubes .
(15)
T he value of
is
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ID : ph-8-Cubes-and-Cube-Root [3]
Answers
(1)
13
16
Step 1
We have been asked to f ind the value of
.
Step 2
Now,
=(
(13 × 13 × 13)
(16 × 16 × 16)
(13)3
=(
)
(16)3
= ((
13
16
=
)3)
)
1
3
1
3
1
3
13
16
Step 3
T heref ore, the value of
is
13
.
16
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ID : ph-8-Cubes-and-Cube-Root [4]
(2)
169 m2
Step 1
Let us assume the side of a cube shape container is a.
Step 2
Volume of cube = 2197 m3
also, volume of the cube = a3
compare both, a3 = 2197
⇒ a = (2197)
1
3
⇒ a = 13
Step 3
Area of side of the cube = a2
= 132
= 169
Step 4
T heref ore, the area of the f ace of cube shape container is 169 m2.
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ID : ph-8-Cubes-and-Cube-Root [5]
(3)
10648
Step 1
T otal population of the town is a perf ect cube.
Number of men in the town = 31207
Number of women in the town = 43329
Step 2
It is given that the number of children in the village is also the perf ect cube, and is more
than 10640. Let us f irst f ind the values of perf ect cubes more than 10640.
Step 3
Cube root of 10640 = 21.994488977725 and the perf ect cubes more than 10640 will be the
cubes of all integers more than 21.994488977725, i.e. 223, 233, 24 3, etc.
Step 4
T he population of children will be the smallest value out of 223, 233, 24 3, etc, f or which the
total population of the village is a perf ect cube.
Step 5
Let us see if the population of the children can be 223 (= 10648). In this case, the total
population of the village will be = 31207 + 43329 + 223
= 31207 + 43329 + 10648
= 85184
Since the cube root of 85184 is 44, which is an integer, the total population of the village in
this case is indeed a perf ect cube.
Step 6
T heref ore, the smallest possible number of children in the village is 223 = 10648.
(4) 50653
Step 1
According to the question, we have to f ind the value of 37 3.
Step 2
Now,
37 3 = 37 × 37 × 37
= 1369 × 37
= 50653
Step 3
T heref ore, the value of 37 3 is 50653.
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ID : ph-8-Cubes-and-Cube-Root [6]
(5)
8
Step 1
According to the question, there are three numbers a, b and c.
b is 4 times a, that is, b = 4a.
c is 0.2 times b, that is, c = 0.2b.
or c = 0.2 × 4a = 0.8a
Step 2
It is also given that sum of their cubes is 524.096,
a3 + b3 + c3 = 524.096
Step 3
Put the value of b and c in above equation.
a3 + (4a)3 + (0.8a)3 = 524.096
⇒ a3(1 + 64 + 0.512) = 524.096
⇒ a3 =
524.096
65.512
⇒ a3 = 8
⇒ a3 = (2)3
⇒a=2
Step 4
Now put value of a in b=4a
⇒b=4 ×2
⇒b=8
Step 5
T heref ore, the value of b is 8.
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ID : ph-8-Cubes-and-Cube-Root [7]
(6) 432
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 that a number is not the perf ect cube.
i. Resolve the given number into prime f actors.
ii. Create group of prime f actors such that each group has three equal f actors (triplets).
iii. If there is something lef t af ter grouping of prime f actors, the number is not the
perf ect cube otherwise perf ect cube.
Step 2
Prime f actors of 1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
T here is no prime f actor lef t af ter creating group of triplets of prime f actors. T heref ore,
1728 is the perf ect cube.
Step 3
Prime f actors of 432 = 2 x 2 x 2 x 2 x 3 x 3 x 3
Af ter creating group of triplets, 2 will remain un-grouped. T heref ore, 432 is not the perf ect
cube.
Step 4
Prime f actors of 2744 = 2 x 2 x 2 x 7 x 7 x 7
T here is no prime f actor lef t af ter creating group of triplets of prime f actors. T heref ore,
2744 is the perf ect cube.
Step 5
Prime f actors of 729 = 3 x 3 x 3 x 3 x 3 x 3
T here is no prime f actor lef t af ter creating group of triplets of prime f actors. T heref ore,
729 is the perf ect cube.
Step 6
T heref ore, 432 is not the perf ect cube.
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ID : ph-8-Cubes-and-Cube-Root [8]
(7)
125
216
Step 1
We have been asked to f ind the value of (
5
)3.
6
Step 2
Now,
(
5
)3=
6
=
53
63
=
5×5×5
6×6×6
125
216
Step 3
T heref ore, the value of (
5
)3 is
6
(8)
125
.
216
216:1
Step 1
If we multiply the number x with 6, the resulting number is: 6x.
Step 2
T he cube of the resulting number 6x = (6x)3
= 216x3
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
=
216x3
x3
=
216
1
Step 5
T heref ore, the ratio of the new number to the cube of the original number is 216:1.
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ID : ph-8-Cubes-and-Cube-Root [9]
(9) 9
Step 1
Let's f irst f ind all prime f actors of 729.
729 = 3 × 3 × 3 × 3 × 3 × 3
Step 2
Now,
= 3√(3 × 3 × 3 × 3 × 3 × 3)
= 3√(33 × 33)
=3×3
=9
(10) d. 4913x3
Step 1
According to the question,
the f irst number is x and the second number is 16 x.
Step 2
Sum of the two numbers = (x + 16 x)
Step 3
Cube of the sum = (x + 16 x)3
= (17 x)3
= 4913 x3
Step 4
T heref ore, the result will be 4913x3.
(11) c. For an integer, the cube root will always be greater than the square root
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ID : ph-8-Cubes-and-Cube-Root [10]
(12) a. -40
Step 1
Let's f irst f ind prime f actors of -400,
-400 = -1 × 2 × 2 × 2 × 2 × 5 × 5
Step 2
Similarly f ind prime f actors of 160,
160 = 2 × 2 × 2 × 2 × 2 × 5
Step 3
Now,
³√(-400) ³√(160)
= ³√(-1 × 2 × 2 × 2 × 2 × 5 × 5) ³√(2 × 2 × 2 × 2 × 2 × 5)
= ³√(-1 × 2 × 2 × 2 × 2 × 5 × 5 × 2 × 2 × 2 × 2 × 2 × 5)
= ³√[-1 × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (5 × 5 × 5)]
= ³√[-13 × 23 × 23 × 23 × 53]
= (-1 × 2 × 2 × 2 × 5)
= -40
(13) d. 32768
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 32768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
T here is no prime f actor lef t with the group of prime f actors. So, 32768 is a perf ect cube.
Step 3
Prime f actor of 15629 = 15629
T here is a prime f actor lef t. So, 15629 is not a perf ect cube.
Step 4
Prime f actors of 26999 = 7 x 7 x 19 x 29
T here is a prime f actor lef t. So, 26999 is not a perf ect cube.
Step 5
Prime f actors of 29792 = 2 x 2 x 2 x 2 x 2 x 7 x 7 x 19
T here is a prime f actor lef t. So, 29792 is not a perf ect cube.
Step 6
T heref ore, 32768 is the perf ect cube.
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ID : ph-8-Cubes-and-Cube-Root [11]
(14)
5824
Step 1
Let us assume that x and y are the two numbers.
Step 2
According to the question, the sum of two numbers is 28, that is,
x + y = 28 ........... (1)
Step 3
And the dif f erence of two numbers is 4, that is,
x - y = 4 ........... (2)
Step 4
On adding equations (1) and (2),
x + y + x - y = 28 + 4
⇒ 2x = 32
⇒ x = 16
Step 5
Now using equation (1),
x + y = 28
⇒ 16 + y = 28
⇒ y = 28 - 16 = 12
Step 6
Now, sum of their cubes = x3 + y3
= (16)3 + (12)3
= 5824
Step 7
T heref ore, the value of the sum of their cubes is 5824.
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ID : ph-8-Cubes-and-Cube-Root [12]
(15)
1.5
Step 1
Let x = ³√(3.375)
Step 2
Multiply both lef t and right hand side by 10,
10x = 10 × ³√(3.375)
⇒ 10x = ³√(3.375 × 1000)
⇒ 10x = ³√(3375)
Step 3
Now f ind the cube root of 3375
⇒ 10x = ³√(3375) = 15
⇒ x = 15/10
⇒ x = 1.5
Step 4
T heref ore, the value of ³√(3.375) is 1.5.
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