1 Class: Subject: Topic: VIII Mathematics Squares and Square Roots The list of Subtopics Why these? Why should I care? Common Mistakes Squares of numbers Square roots Knowing where to use squares and square roots Properties of square numbers Patterns of a square of a number Finding the square of a number Finding square roots Square roots of a decimal Estimating square roots Unable to differentiate between problems that ask for squares versus those that ask for square roots Placing bars over numbers for finding square root through long division method Application of squares and square roots is used in a wide range of topics from mensuration and trigonom- etry to calculus
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Class: Subject: Topic: VIII Mathematics Squares and Square Roots
The list of Subtopics Why these? Why should I care? Common Mistakes
Squares of numbers
Square rootsKnowing where to usesquares and square roots
Properties of squarenumbers
Patterns of a square of anumber
Finding the square of anumber
Finding square roots
Square roots of a decimal
Estimating square roots
Unable to differentiatebetween problems that askfor squares versus those thatask for square roots
Placing bars over numbersfor finding square rootthrough long divisionmethod
Application of squares andsquare roots is used in awide range of topics frommensuration and trigonom-etry to calculus
8. How many numbers lie between squares of the following numbers?
(i) 12 and 13 (ii) 25 and 26 (iii) 99 and 100
9. Is 63504 a perfect square? If so, find the number whose square is 63504.
10. Determine whether squares of the following numbers are even or odd.
i) 213 ii) 3824 iii) 9777 iv) 40028
11. Write a Pythagorean triplet whose one member is
(i) 6 (ii) 14 (iii) 16 (iv) 18
12. Find the square of the following numbers
(i) 32 (ii) 35 (iii) 86 (iv) 93 (v) 71 (vi) 46
13. If 8 is one of the numbers in a Pythagorean triplet, then find the triplet.
12
14. Complete the following table of perfect squares and their square roots:
Perfect Square Square Root
1 1
4 2
9 3
36
8
100
13
Square roots
Let us consider 811/2
As 81 is raised to the power 1/2, it is said to be the square root of 81.
Denoted as 12
Exponential Radical form form
81 81
Thus, any number a raised to the power 1/2 is said to be the square root of the givennumber, denoted as a .
Finding Square root of a Number by Prime Factorization Method
In the previous topics we have seen different methods of finding the squares of numbersgiven. Let us now have a look at the various methods for finding the square root of thegiven number.
Step-1: Write the prime factorisation of the given number.
Step-2: Pair the factors such that primes in each pair are equal.
* * *
13
Step-3: Choose one prime from each pair and multiply all such primes.
Step-4: The product thus obtained is the square root of the given number.
Let us understand this method through the example given below.
Example: Find the square root of 24336.
24336 (2 2) (2 2) (3 3) (13 13)
= 2 × 2 × 3 × 13 = 156
24336 156
WORKSHEET 2
1. What could be the possible ‘one’s’ digits of the square root of each of the following
numbers?
(i) 9801 (ii) 99856 (iii) 998001 (iv) 657666025
2. Without doing any calculation, find the numbers which are surely not perfect
squares.
(i) 153 (ii) 257 (iii) 408 (iv) 441
3. Find the square roots of 100 and 169 by the method of repeated subtraction.
4. Find the square roots of the following numbers by the pme factorisation method.
7. The students of Class VIII of a school donated Rs 2401 in all, for Prime Minister’s
National Relief Fund. Each student donated as many rupees as the number ofstudents in the class. Find the number of students in the class.
8. 2025 plants are to be planted in a garden in such a way that each row
contains as many plants as the number of rows. Find the number of rows and thenumber of plants in each row.
9. Find the smallest square number that is divisible by each of the numbers 4, 9,
and 10.
10. Find the smallest square number that is divisible by each of the numbers 8, 15,
and 20.
11. A square has an area of 16 units2.
a. What is the side length of a square of this area?b. Draw a square of area 16 units2 below.c. What is the square root of 16?d. Explain why your answers in parts (a) and (c) are the same.
12. A checkerboard is a square made up of 32 black and 32 red squares.
Assume that each square has a side length of 1 unit.a. What is the total area of the checkerboard?b. What is the side length of the checkerboard?c. Explain how your answers in parts (a) and (b) help you determine the
square root of 64.
13. The symbol means the positive or principal square root of a number.
a. Evaluate 121 .
b. What is the negative square root of 121?
c. A square has an area of 121 units2. What is the side length of square ofthis area?
d. Explanin why the answer to part (c) can only be the positive square rootof 121.
15
Long Division Method:
Let us understand this method through the following example.
Example: Find the square root of 467856.
Step-1: Placing bars over every pair of digits i.e., 4678 56 .
Step-2: Finding the largest number whose square is less than or equal to the left most bar.
i.e., 36 < 46 6 is the required divisor. Now divide 46 by 6.
4678 5636
10
6
6
Step-3: Bring down the number under the next bar, beside the remainder of Step 2.
4678 563610 78
6
6
Step-4: Doubling the quotient i.e., 2 × 6 = 12.
46 78 5636
10 78
68
6
10 2454
128
Step-5: Guessing the largest possible digit to be taken beside the divisor and also to get thenew digit in the quotient. The required digit is8. Now divide 1078 by 128 and get theremainder 54.
Step-6: Bringing down the number under the next bar to the right of the new remainder.
4678 56361078
68
6
102454 56
128
16
Step-7: Repeating the steps 4, 5 and 6 till all the bars have been considered. The finalquotient is the required square root.
46 78 5636
1078
684
6
102454 56
128
136454 56
0
467856 684
Note: This method is more efficient with larger numbers, but it can also be used to findsquare root of smaller numbers i.e., 3 digit or 4 digit numbers.
Estimating Square Roots
Most calculators have a square root button that quickly calculates square roots. There areother ways of calculating square roots but they aren’t quick. If you need an approximatevalue for a square root you can use a method like the one below.
What is the square root of 42?
What two squares does 42 come between? 36 and 49
What are the square roots of 36 and 49 6 and 7
So the square root of 42 is between 6 and 7.Let’s use some trial and error to get anapproximate answer.
Let’s try 6.5 42 ÷ 6.5 = 6.46
We’re looking to get the number we divide by to be as close to the answer we get aspossible. In this case we have 6.5 and 6.46Close, but we can get closer.
Let’s take the average of 6.5 and 6.46 and try that. 42 ÷ 6.48 = 6.48
So we have an square root that is accurate to two decimal places. You can repeatthese steps to get as accurate an answer as you want.
17
Finding Square Root by Assumption Method
Steps for finding the square roots up to four digit numbers, without using eitherfactorization or division methods.
Step-1: Find the largest number whose square is less than (or) equal to the numberunder the left most bar. This is the tens digit of the square root.
Step-2: Find the units digit by squaring the relative number.
Step-3: Choose the correct digit by squaring one possible square root and comparing itwith the given number.
Example: Find the square root of 9801.
Sol: 98 01
Step-1: 92 = 81 is the largest square number < 98;
The tens digit in the square root of 9801 is 9.
9801 9 ? ;
Step-2: 12 = 1 ; 92 = 8 1 in both the cases, the units digit is 1.
Step-3: Trial 912 = 8281 9801
i.e., 1 doesn’t satisfy the units digit of square root of 9801.
9801 99
Example: Find the square roots of 144 and 6561.
Sol: 144
Step-1: 1 44
12 = 1
Tens digit in square root of 144 is 1. 144 1 ?
Step-2: 22 = 4 ; 82 = 6 4
In both the cases, the units digit is 4.
Step-3: Trial 122 = 144 144 12
Example: Find the square roots of 6561.
18
Sol: 6561
Step-1: 65 61
82 = 64
Tens digit in square root of 6561 is 8 . 6561 8 ?
Step-2: 12 = 1 ; 92 = 8 1
In both the cases, the units digit is 1.
Step-3: Trial 802 = 6400 is closer to 6561 than 902 = 8100
6561 is closer to 80 than 90
The tens digit of square root may be 8
Square root of 6561 may be 81
Verification : 81×81=6561
6561 81
Square roots of decimals
Step-1: Place bars on the integral part of the number in the usual manner.
Step-2: Place bars on the decimal part on every pair of digits beginning with the firstdecimal place.
Step-3: Start finding the square root by the division process as usual.
Step-4: Place decimal point in the quotient as soon as the integral part is over.
Step-5: Stop when the remainder is zero.
Step-6: The quotient at this stage is the square root.
Example: Find the square root of 0.00059049
Step-1: Here, integral part is 0. So place a bar on 0.
Step-2: Starting from the first decimal point, place the bars on every pair of digits. i.e.,
00059049 .
19
Step-3: Start finding the square root and place the decimal point in the quotient assoon as the integral part is over.
0.00 0590494
1 90
0.0243
2
44
0
1 76
14491449
483
0.0005905 0.0243
Tip (i) : If the integral part is zero in the given decimal number, then the integral part inthe square root is also zero.
Tip (ii): If the first pair after the decimal point is 00, then the first digit after the decimalin the square root is also zero.
20
WORKSHEET 3
1. A man arranges 15376 apple plants in his garden in an order, so that there are as
many rows as there are apple plants in each row. Find the number of rows.
2. The product of three consecutive even numbers when divided by 8 is 720. Find
the product of their square roots.
3. What least number must be subtracted from 16160 to get a perfect square? Also
find the square root of this perfect square.
4. What least number must be added to 2945 to get a perfect square number? What
is the resulting number? Find the square root of the resulting number.
5. Find the greatest number of 5 digit which is a perfect square.
6. Find the least number of six digits which is a perfect square.
7. Find the value of 0.9 up to 3 places of decimal.
8. Find the value of 3 correct to 3 places of decimal.
9. Find the value of 37 up to four decimal places
10. Find the square root of each of the following numbers by division method.