Download Grade 8 Square and Square Roots

ID : pk-8-Square-and-Square-Roots [1]
Grade 8
Square and Square Roots
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Answer t he quest ions
(1)
If you have a square ground with an area of 30
7
square metres, what would be the length
28
of each side?
(2)
Solve f or x
(3)
Find the value of the f ollowing expression:
√ 1 + 3 + 5 + 7 +...+ 31
(4) What is the square root of 1296?
(5)
Which of the f ollowing numbers cannot be the area of a square that has an integer value side?
1089 2601 840 841
(6) Students f rom grade 3 class decided to collect money f or earthquake relief . T he amount (in Rs.)
collected by each students equals to number of students in the class, If they collected a total of
Rs.4761, how many students were there in the class?
(7) Find the smallest even number whose square is larger than 1548?
(8)
Which is the smallest number that can be multiplied to 19800 to give a perf ect square?
Choose correct answer(s) f rom given choice
(9) A number n is called a square number if there exists:
a. a natural number m such that n = m + m
b. a natural number m such that m = n x n
c. a natural number m such that n = m x m
d. a natural number m such that m = n + n
(10) Which is the smallest number that can be used to divide 622908 to give a perf ect square?
a. 146
b. 143
c. 11
d. 13
(11) For any given number x, which of the f ollowing can be pythagorean triplets?
a. 2m, m2-1,m2
b. 2m+1, m2-1,m2+1
c. 2m-1, m2-1,m2
d. 2m, m2-1,m2+1
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ID : pk-8-Square-and-Square-Roots [2]
Fill in t he blanks
(12)
(13)
(14)
If the square root of 2 is 1.41, then the square root of 1058 is
T he largest perf ect square number that is smaller than the largest 3 digit number is
Find smallest possible value of
=
(15) Every day, a f actory making bats sends 1464 bats to the town market and 3177 bats to markets
in other states. We know that a little more than 683 bats are remaining in the f actory. If we know
that the f actory makes a perf ect square number of bats every day, then the smallest possible
number of bats remaining in the f actory is
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ID : pk-8-Square-and-Square-Roots [3]
Answers
(1)
11
m
2
Step 1
Let's assume that the length of a side of given square ground is a.
Area of square = 30
=
847
7
square metres
28
square metres
28
Step 2
Area of square = (a)2
a2 =
847
28
⇒a=
√847
√28
⇒a=
√(11 × 11 × 7)
√(2 × 2 × 7)
⇒a=
√(11 × 11 × 7)
√(2 × 2 × 7)
⇒a=
√(11)2
√(2)2
⇒a=
11
2
Step 3
T heref ore, the length of each side of the square ground is
11
m.
2
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ID : pk-8-Square-and-Square-Roots [4]
(2)
90
Step 1
We have to f ind the value of x f rom the f ollowing,
.
Step 2
Squaring both side
⇒
x
⇒
=
540
15
x
⇒ x2 = 540 × 15
⇒ x2 = ( 2 × 2 × 3 × 3 × 3 × 5) × (3 × 5)
⇒
⇒
⇒
x2 = (2 × 2) × (3 × 3) × (3 × 3) × (5 × 5)
x = ± (2 × 3 × 3 × 5)
x = ±90
Step 3
If x is negative, expression in question will result in square root of a negative number which
is not possible. T heref ore, the value of x is 90.
(3)
±16
Step 1
We have been asked to f ind the value of √ 1 + 3 + 5 + 7 +...+ 31 .
Step 2
T his is an arithmetic sequence and the sum of the f irst n terms is given by:
Sn =
n(a1 + an)
2
where, n = number of terms, a1 = f irst term, an = last term
S16 = √(
= √(
16(1 + 31)
512
)
2
)
2
= √ 256
= ±16
Step 3
T heref ore, the value of √ 1 + 3 + 5 + 7 +...+ 31 is +16 or -16
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ID : pk-8-Square-and-Square-Roots [5]
(4) 36
Step 1
We have been asked to f ind the square root of 1296.
Step 2
Prime f actors of 1296 are,
2 × 2 × 2 × 2 × 3 × 3 × 3 × 3.
Step 3
Now, make pairs of prime f actors.
(2 × 2) × (2 × 2) × (3 × 3) × (3 × 3)
Step 4
T here is no prime f actor lef t which is not part of a pair. Square root is given by the
multiplication of e element f rom each pair of prime f actors. i.e.
2×2×3×3
= 36
Step 5
T heref ore, the square root of 1296 is 36.
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ID : pk-8-Square-and-Square-Roots [6]
(5)
840
Step 1
We know the area of square = side2
Since it is given that side has an integer value, the area must be a perf ect square.
Let's check if given areas are perf ect square or not
Step 2
Let's check if 1st area of 1089 is perf ect square or not using division method,
33
3 10 89
+3 9
63
189
+3
189
0
1089 is a perf ect square, theref ore it could be the are of given square
Step 3
Let's check if 2nd area of 2601 is perf ect square or not using division method,
51
5 26 01
+ 5 25
101
101
+1
101
0
2601 is a perf ect square, theref ore it could be the are of given square
Step 4
Let's check if 3rd area of 840 is perf ect square or not using division method,
28.9
2 8 40 . 00
+2 4
48
440
+8
384
569
5600
+9
5121
479
840 is not a perf ect square, theref ore it cannot be the are of given square
Step 5
Let's check if 4 th area of 841 is perf ect square or not using division method,
29
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ID : pk-8-Square-and-Square-Roots [7]
2 8 41
+2 4
49
441
+9
441
0
841 is a perf ect square, theref ore it could be the are of given square
Step 6
T heref ore, 840 cannot be the area of given square.
(6) 69
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 Rs.x.
So, total collected money = x2.
Step 3
Collected money by the students = Rs.4761
x2 = 4761
⇒ x = √4761
⇒ x = 69 (ignore negative since number of students cannot be negative)
Step 4
T heref ore, there were 69 students in the class.
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ID : pk-8-Square-and-Square-Roots [8]
(7) 40
Step 1
We have been asked to f ind the smallest even number whose square is larger than 1548.
Step 2
First of all we have to f ind the square root of 1548.
39.3
3 15 48 . 00
+3 9
69
648
+9
621
783
2700
+3
2349
351
Step 3
We can notice that square root of 1548 is greater than 39 and less than 40.
Step 4
T heref ore, the smallest even number whose square is larger than 1548, will be an even
number which is larger than 39. T hus, answer is 40.
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ID : pk-8-Square-and-Square-Roots [9]
(8)
22
Step 1
Perf ect square is a number made by squaring a whole number. Steps to check if a number is
perf ect square are as f ollows.
i. Resolve the given number into prime f actors.
ii. Make pairs of prime f actors such that both the f actors in each pair are equal.
iii. If there is nothing lef t without the pair of prime numbers so the number is perf ect
square otherwise not.
Step 2
Prime f actors of 19800 = 2 × 2 × 2 × 3 × 3 × 5 × 5 × 11
Step 3
Lets' create pairs of prime f actors,
19800 = (2 × 2) × (3 × 3) × (5 × 5) × 2 × 11
Step 4
T his number is not the perf ect square, since all f actors are not paired. If we multiply this
number with remaining prime f actors (i.e. 2 × 11 = 22), all prime f actors will be paired and
number will be perf ect square.
Step 5
T heref ore, 22 is the smallest number that can be multiplied to 19800 to give the perf ect
square.
(9) c. a natural number m such that n = m x m
Step 1
Square number: T he number we get af ter multiplying an integer (not a f raction) by itself .
f or example: 3 × 3 = 9, so 9 is a square number.
Step 2
Condition for square number:
A number n is called a square number if there exists a natural number m such that n = m ×
m.
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ID : pk-8-Square-and-Square-Roots [10]
(10) b. 143
Step 1
Perf ect square is a number made by squaring a whole number. Steps to check if a number is
perf ect square are as f ollows.
i. Resolve the given number into prime f actors.
ii. Make pairs of prime f actors such that both the f actors in each pair are equal.
iii. If there is nothing lef t without the pair of prime numbers so the number is perf ect
square otherwise not.
Step 2
Prime f actors of 4356 = 2 × 2 × 3 × 3 × 11 × 11
Step 3
Lets' create pairs of prime f actors,
4356 = (2 × 2) × (3 × 3) × (11 × 11) × 11 × 13
Step 4
T his number is not the perf ect square, since all f actors are not paired. If we divide this
number by remaining prime f actors (i.e. 11 × 13 = 143), all prime f actors will be paired and
number will be perf ect square.
Step 5
T heref ore, 143 is the smallest number that can be used to divide 622908 to give the
perf ect square.
(11) d. 2m, m2-1,m2+1
(12)
32.43
Step 1
We have been asked to f ind the square root of 1058 if square root of 2 is 1.41.
Step 2
Now
√(1058) = √(529 × 2)
=√((23)2 × 2)
= 23 × √(2) = 23 × 1.41
= 32.43
Step 3
T heref ore, square root of 1058 is 32.43 if square root of 2 is 1.41.
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ID : pk-8-Square-and-Square-Roots [11]
(13)
961
Step 1
T he largest 3 digit number is 999. We need to f ind the largest perf ect square number that is
smaller than 999.
Step 2
Let us f irst look at the square root of 999. T he square root of 999 = 31.60696
T he largest perf ect square number smaller than 999 must be the square of the integer just
smaller than 31.60696.
Step 3
T he integer just smaller than 31.60696 is 31 and the square of 31 = (31)2
= 961
Step 4
T heref ore, the largest perf ect square number that is smaller than the largest 3 digit number
is 961.
(14)
-26
Step 1
Lets assume smallest possible value of expression is S.
S=
⇒ S = (±9) + (±17)
Step 2
S will be smaller if f irst number is negative,
⇒ S = (-9) + (±17)
Step 3
Similarly, S will be smaller if second number is negative,
⇒ S = (-9) + (-17)
Step 4
T hus, smallest possible value is -26
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ID : pk-8-Square-and-Square-Roots [12]
(15)
688
Step 1
Number of bats sent to the town market = 1464 bats
Number of bats sent to the markets in other states = 3177 bats
Number of bats remaining in the f actory = little more than 683 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 1464 + 3177 + 683
= Little more than 5324
Step 3
T hus the total number of bats made by the f actory every day is little more than 5324 and
such that the number is a perf ect square.
Step 4
Let's f ind the next number greater than 5324 that is a perf ect square. For that, let us f irst
look at the square root of 5324:
Square root of 5324 = 72.965745387819
T he number next to 5324 that is a perf ect square will be the square of the integer just more
than the square root 72.965745387819: that is 73.
Step 5
Hence the number of bats made in the f actory every day = 732 = 5329.
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
= 5329 - 1464 - 3177
= 688
Step 7
T heref ore, the number of bats remaining in the f actory is 688.
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