Download Grade 8 Square and Square Roots

ID : cn-8-Square-and-Square-Roots [1]
Grade 8
Square and Square Roots
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Answer t he quest ions
(1)
Students f rom grade 7 class decided to collect money f or earthquake relief . T he amount (in ¥)
collected by each students equals to number of students in the class, If they collected a total of
¥3600, how many students were there in the class?
(2)
T here are 3126 soldiers in a platoon. T he commander of the platoon wants to impress the
visiting general by arranging his soldiers in such a manner that they are arranged in n rows, and
each row has n soldiers. He f inds that he cannot do this with the number of soldiers he has.
What is the minimum number of soldiers he needs additionally to do this kind of a f ormation?
(3)
What is the expansion of (a + b)2?
(4) Find the smallest even number whose square is larger than 2120?
(5)
What is the smallest number that must be subtracted f rom 393 to get a perf ect square?
Choose correct answer(s) f rom given choice
(6) If you subtract a number x f rom 42 times that number, and then take the square of the
dif f erence, what will the result be?
a. 43x2
b. 39x2
c. 1681x2
d. 42x2
(7) Every day, a f actory making bats sends 2206 bats to the town market and 1429 bats to markets
in other states. We know that a little more than 981 bats are remaining in the f actory. If we know
that the f actory makes a perf ect square number of bats every day, what is the smallest possible
number of bats remaining in the f actory?
(8)
a. 993
b. 972
c. 989
d. 988
For any given number x, which of the f ollowing can be pythagorean triplets?
a. 2m-1, m2-1,m2
b. 2m, m2-1,m2
c. 2m, m2,m2+1
d. 2m, m2-1,m2+1
(9) What is the expansion of (a - b)2?
a. a2 - 2ab + b 2
b. a2 + ab + b 2
c. a2 - ab + b 2
d. a2 + 2ab + b
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ID : cn-8-Square-and-Square-Roots [2]
(10) What is the sum of the f irst 177 odd numbers?
a. 31354
b. 31329
c. 31152
d. 30976
(11) What is the value of
a.
?
184
b.
23
529
8
c.
184
64
d.
529
529
(12) T here are two numbers such that the dif f erence of their squares is 60 and sum of the numbers
is 30. Find their dif f erence.
a. 2
b. 7
c. 3
d. 1
Fill in t he blanks
(13) T he smallest perf ect square number that is larger than the largest 6 digit number is
(14) T he total population of a village is a perf ect square. T he number of men in the village is 31006,
and the number of women in the village is 39779.
If we know that the number of children in the village is also a perf ect square and is more than
17420, then the smallest possible number of children in the village is
(15)
If x = 1.59, then the value of
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is
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ID : cn-8-Square-and-Square-Roots [3]
Answers
(1)
60
Step 1
Let us assume the number of students is x.
Step 2
According to the question, if there are x students and each student collected ¥x.
So, total collected money = x2.
Step 3
Collected money by the students = ¥3600
x2 = 3600
⇒ x = √3600
⇒ x = 60 (ignore negative since number of students cannot be negative)
Step 4
T heref ore, there were 60 students in the class.
(2)
10
Step 1
It is given that a platoon has 3126 soldiers.T he number of rows and the soldiers in each
row is the same.
Step 2
We have to f ind the number of more soldiers that should be there, so that when the
commander arranged them, the number of rows and the number of soldiers in each row are
same.
T hat is, the number which should be added to 3126 to make it a perf ect square has to be
calculated.
If the square root of 3126 is calculated then the quotient and the remainder is 55 , 101
respectively.It represents that the square of 55 with the sum of 101 is equal to the 3126.
T he next number is 56 and 562 = 3136
Hence the number to be added to 3126 to make it a perf ect square
= 562 - 3126 = 3136 - 3126 = 10
Step 3
T heref ore, required number of soldiers is 10.
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ID : cn-8-Square-and-Square-Roots [4]
(3)
a2 + 2ab + b 2
Step 1
If you look at the question caref ully, you will notice that you have to f ind out the expansion
of (a + b)2 .
Step 2
Now,
(a + b)2 = (a + b)(a + b)
= (a + b)a + (a + b)b
= a2 + ab + ab + b2
= a2 + 2ab + b 2
Step 3
T heref ore the expansion of (a + b)2 is a2 + 2ab + b 2 .
(4) 48
Step 1
We have been asked to f ind the smallest even number whose square is larger than 2120.
Step 2
First of all we have to f ind the square root of 2120.
46.0
4 21 20
+ 4 16
86
520
+6
516
4
Step 3
We can notice that square root of 2120 is greater than 46 and less than 47.
Step 4
T heref ore, the smallest even number whose square is larger than 2120, will be an even
number which is larger than 46. T hus, answer is 48.
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ID : cn-8-Square-and-Square-Roots [5]
(5)
32
Step 1
Let's f irst f ind the square root of 393 using division method,
19.8
1 3 93 . 00
+1 1
29
293
+9
261
388
3200
+8
3104
96
T his shows that square root of 393 is between 19 and 20
Step 2
Now, square of 19,
192 = 19 × 19 = 361
Step 3
T heref ore, number that must be subtracted f rom 393, to get perf ect square,
= 393 - 361
= 32
(6) c. 1681x2
Step 1
According to the question,
f irst number is x and
second number is 42 x.
Step 2
Dif f erence of two numbers = (42 x - x)
Square of the dif f erence = (42 x - x)2
= (41 x)2
= 1681 x2
Step 3
T heref ore the result will be 1681x2.
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ID : cn-8-Square-and-Square-Roots [6]
(7) c. 989
Step 1
Number of bats sent to the town market = 2206 bats
Number of bats sent to the markets in other states = 1429 bats
Number of bats remaining in the f actory = little more than 981 bats
Step 2
According to the question, the f actory makes a perf ect square number of bats every day.
T his means the total number of bats in the f actory must be a perf ect square.
T otal number of bats = Little more than the sum 2206 + 1429 + 981
= Little more than 4616
Step 3
T hus the total number of bats made by the f actory every day is little more than 4616 and
such that the number is a perf ect square.
Step 4
Let's f ind the next number greater than 4616 that is a perf ect square. For that, let us f irst
look at the square root of 4616:
Square root of 4616 = 67.941151005852
T he number next to 4616 that is a perf ect square will be the square of the integer just more
than the square root 67.941151005852: that is 68.
Step 5
Hence the number of bats made in the f actory every day = 682 = 4624.
Step 6
T he number of bats remaining in the f actory every day = T otal number of bats made every
day - T otal number of bats sent to markets
= T otal number of bats made every day - total number of bats sent to the local market total number of bats sent to the markets in other states
= 4624 - 2206 - 1429
= 989
Step 7
T heref ore, the number of bats remaining in the f actory is 989.
(8)
d. 2m, m2-1,m2+1
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ID : cn-8-Square-and-Square-Roots [7]
(9) a. a2 - 2ab + b 2
Step 1
If you look at the question caref ully, you will notice that you have to f ind out the expansion
of (a - b)2 .
Step 2
Now,
(a - b)2 = (a - b)(a - b)
= (a - b)a - (a - b)b
= a2 -ab - ab - (-b)b
= a2 - 2ab + (b)b
= a2 - 2ab + b 2
Step 3
T heref ore the expansion of (a - b)2 is a2 - 2ab + b 2 .
(10) b. 31329
Step 1
If you look at the question caref ully, you will notice that you have to f ind out the value of
the sum of the f irst 177 odd numbers.
Step 2
Sum of f irst n odd numbers = n 2
(where, n = 177)
= (177)2
= 31329
Step 3
T heref ore the sum of f irst 177 odd numbers is 31329.
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ID : cn-8-Square-and-Square-Roots [8]
(11)
64
d.
529
Step 1
We have been asked to f ind the value of
.
Step 2
Now,
=
8×8
23 × 23
=
64
529
Step 3
T heref ore, the value of
is
64
.
529
(12) a. 2
Step 1
Let us assume x and y are the two numbers.
Step 2
It is given that, the dif f erence of their squares is 60, that is, x2 - y2 = 60 ----(1)
Step 3
It is also given that sum of the numbers is 30, that is, x + y = 30 ----(2)
Step 4
We know that,
x2 - y2 = (x + y)(x - y)
Step 5
On substituting values f rom Eq. (1) and (2),
60 = 30 × (x - y)
x - y = 60/30
x- y=2
Step 6
T heref ore, their dif f erence is 2.
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ID : cn-8-Square-and-Square-Roots [9]
(13)
1000000
Step 1
T he largest 6 digit number is 999999. We need to f ind the smallest perf ect square number
that is larger than 999999.
Step 2
Let us f irst look at the square root of 999999. T he square root of 999999 = 999.9995
T he smallest perf ect square number larger than 999999 must be the square of the integer
just larger than 999.9995.
Step 3
T he integer just larger than 999.9995 is 1000 and the square of 1000 = (1000)2
= 1000000
Step 4
T heref ore, the smallest perf ect square number that is larger than the largest 6 digit number
is 1000000.
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ID : cn-8-Square-and-Square-Roots [10]
(14)
17424
Step 1
T otal population of the village is a perf ect square.
Number of men in the village = 31006
Number of women in the village = 39779
Step 2
It is given that the number of children in the village is also a perf ect square, and is more
than 17420. Let us f irst f ind the values of perf ect squares more than 17420.
Step 3
Square root of 17420 = 131.98484761517 and the perf ect squares more than 17420 will
be the squares of all integers more than 131.98484761517, ie 1322, 1332, 134 2, etc.
Step 4
T he population of children will be the smallest value out of 1322, 1332, 134 2, etc, f or which
the total population of the village is a perf ect square.
Step 5
Let us see if the population of the children can be 1322 (= 17424). In this case, the total
population of the village = 31006 + 39779 + 1322
= 31006 + 39779 + 17424
= 88209
Since the square root of 88209 is 297, which is an integer, the total population of the
village in this case is indeed a perf ect square.
Step 6
T heref ore, the smallest possible number of children in the village is 1322 = 17424.
(15)
3.77
Step 1
Using algebraic identity (a - b)2 = a2 - 2ab + b2,
=
⇒ = (x - 1) + 2x
⇒ = 3x - 1
Step 2
Now replace x by 1.59,
⇒ = (3×1.59 - 1)
⇒ = 3.77
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