SQUARE AND SQUARE ROOT CLASS 8(EXERCISE6.1) | NCERT | PART 2
SQUARE AND SQUARE ROOT CLASS 8(EXERCISE6.1) | NCERT | PART 2
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00:00Hi everyone, welcome to Insightful Maths.
00:09This is session no.
00:102 of square and square root topic that we have done in the previous class.
00:15So I hope you have watched that video.
00:18If not, please watch that video and then start this one because this is the continuation
00:23of what we have already done.
00:26If you have not liked, subscribed or followed my channel, please do the same.
00:31Okay.
00:32So let's continue.
00:33Then in the previous session, we have already understood all these properties of the square
00:39numbers.
00:40That means the square number can be written as the odd square number can be written as
00:45the sum of two consecutive natural numbers or the product of two consecutive odd and
00:51even natural numbers can also be determined using this property, the sum of two, the difference
00:57of two and the square minus one.
01:00If not, please go through that video.
01:03So I'm continuing with a new topic, which is Pythagorean triplet.
01:07You must have studied in grade seven that in a right angle triangle, I'm giving you
01:12a recap.
01:13If I have a right angle triangle, let's say A, B and C, okay?
01:19These two are the arms of this right angle triangle and the longest side, which is opposite
01:26to the right angle triangle.
01:28This is known as your hypotenuse.
01:31So what does that property says?
01:33This longest side hypotenuse square is equal to the sum.
01:40Sum means addition.
01:41Sum of what?
01:42The sum of two numbers.
01:43Which two numbers?
01:44The square of the base, that is A, B square and this vertical height or the perpendicular
01:51A, C square.
01:53That was Pythagoras theorem, okay?
01:56Pythagorean triplet, if we talk about triplet means the group of three numbers.
02:01So Pythagorean triplet should be a triplet or a group of three numbers, which follows
02:06Pythagoras theorem.
02:08So if any number is being given, if any triplet is given, let's say three, four, five and
02:13the question says, is it a Pythagorean triplet?
02:16So what you're going to check, the longest side is always the hypotenuse.
02:22It will not be told to you that this is the hypotenuse, this is base and this is height.
02:27It is but obvious.
02:28The longest side will always be your hypotenuse.
02:32So you need to check longest side square should be equal to the sum of the square of rest
02:40two numbers.
02:41So five square should be equal to the sum of three square and four square.
02:48Five square is 25, three square is nine and four square means four into four, that is
02:5416.
02:55Nine plus 16 is 25 and left hand side is 25.
03:00Are we getting both side equal?
03:02Yes.
03:03So your answer will be yes, it is a Pythagorean triplet.
03:07This is one category of question.
03:09Another category of question in grade 8 now that you will be getting is one number of
03:15the Pythagorean triplet will be given to you and it will be specified that it is just a
03:20one number or the biggest number or the smallest number.
03:24So you have to read the question carefully and then attempt it.
03:28Okay.
03:29So you need to find another two numbers of the triplet.
03:32Here the question is, you have to find a Pythagorean triplet whose the smallest member is eight.
03:40Rest two numbers should be greater than eight.
03:42This is the condition.
03:43There might be triplets wherein eight can be combined with other numbers, but the number
03:49may be lesser than eight as well.
03:51We don't want to take that.
03:53The smallest member has to be eight.
03:55You need to remember the formula for the three numbers.
03:59twice of m, m square minus one and m square plus one.
04:08For any value of m, twice of m, m square minus one, m square plus one.
04:15This gives you your Pythagorean triplet.
04:18This you need to remember.
04:19So given number eight, it should be equal to one of these numbers.
04:25Okay.
04:26It is just hidden triangle.
04:27So I need to put eight equal to any one of these easiest way here, which I can make out
04:33eight is an even number.
04:35So if I put two m equal to eight, I can easily get the value of eight.
04:40So twice of m is equal to eight linear equations in one variable, two is multiplied here.
04:48So on another side, it will divide it and what is the value of m I'm getting?
04:55It is four.
04:57Understand one thing.
04:58One of the numbers twice of m, it is already given eight ticket.
05:03This number we have just substituted equal to eight.
05:07One number is already there.
05:09You need to find other two numbers.
05:11You have to get the value of m.
05:15We have calculated the value of m, which is equal to four, which is the second number.
05:22Now as per the formula, it is m square minus one, m is four.
05:27So it is four square minus one, four square is four into four, that is 16 minus one, it
05:35is 50.
05:36Understood?
05:37Third number as per the formula is m square plus one, which is four square plus one, 16
05:47plus one, it is equal to 17.
05:49So you have got all the three numbers, one, eight was already given, m square minus one,
05:56you have got 15, m square plus one is 17.
06:00Question said that eight has to be the smallest member.
06:03So out of this triplet, triplet it has to be, answer has to be written in like this,
06:09eight comma seven, 15 comma 17, is eight the smallest number?
06:14Answer is yes.
06:15If you put something else between these two, you will get another triplet, but the numbers
06:21can be lesser than eight.
06:23So that means you need to try with some other combination.
06:26It goes like this.
06:27Okay?
06:28I hope this much is understand.
06:30You can re-watch it once again, if you have any doubt.
06:342m, m square minus one, m square plus one, one of the numbers equal to eight, it is already
06:41given.
06:42Find the value of m and substitute here, you will get other two members.
06:47Coming to the next topic now.
06:50In this video, we'll be covering the first exercise of square and square root as well.
06:55Okay.
06:56So next topic is finding the square without multiplication, square kya hota hai?
07:01For example, four square.
07:03So what do we do?
07:04Four into four, which is 60.
07:06It is a one digit number.
07:07So finding the square in this manner is easy.
07:11Another way out, if you're having two or three digit number, if I have to find 15 square
07:17without multiplying 15 into 15, multiplied bina kare, some other method has to be used
07:24for finding.
07:25So what you are going to do, you have to split this 15 in two parts, advisable one part should
07:31have zeros, 10, 100, 1000, it has to be like this.
07:36So I can write 15 as 10 plus 5, iska square.
07:41This much is okay.
07:4210 plus 5 for square, the rule says, first number square, pehle aap first number ka square
07:49lo, plus twice of first number into second number, first number into second number, plus
08:00the square of second number.
08:02What we are doing?
08:04First number ka square, second number ka square, please have a look, twice of first number
08:10and second number.
08:11So what is 10 square?
08:13It is 100 plus 5 into 2 is 10, 10 into 10 is 100, plus 5 into 5 is 25, 100 plus 100
08:25is 200, and 200 plus 25 is 225, that's your answer.
08:31If you calculate 15 into 15, you will again be getting the same answer.
08:36So the rule here or the formula here is, if you have two numbers a and b, so a plus b
08:43ka square, what we have done?
08:45First number ka square, plus second number ka square, plus twice of first number and
08:53second number.
08:54This holds true for any combination of number.
08:57You can try finding the square of 13 using the same property.
09:03You can split like 10 plus 3 ka whole square, then you will do 10 square, twice of first
09:10number, second number, plus second number square, chheke, calculate it yourself and
09:17recheck your answer.
09:20Another now, finding the square root by prime factorization.
09:24Normally we use this method for finding the square root of the number.
09:29Square root means the reverse of square.
09:32If I say 2 square is 4, iska matlab, under root of 4, going back, under root of 4 is
09:39equal to 2.
09:41If 3 square is equal to 9, that means if I go back, under root of 9 is equal to 3.
09:48Theek hai, it goes like this.
09:50For smaller numbers, we can do that orally.
09:59For the bigger numbers, we need to have a rule.
10:02Let's find it here.
10:04Prime factorization means writing the number as the product of its prime factors.
10:0936, it is divisible by 2.
10:122 into 1 is 2.
10:14To 8 is 16.
10:162 into 9 is 18 and 9 can be written as 3 into 3, okay.
10:22So 36, we are finding the square root of 36.
10:27This is the symbol of square root like this.
10:30So under the square root, 36 can be written as 2, 2, 3 and 3, the product of these numbers
10:372, 2, 3 and 3.
10:41Divide the root, make the pairs of similar number, 2 and 2 similar, 1 pair, 3 and 3 similar,
10:491 pair.
10:50Out of these similar numbers, 1, 1 number will come out and rest everything will vanish.
10:57In dono 2 mese, 1, 2 out, out of these 3, 1, 3 out, take the product and your answer
11:04is 6.
11:05Yes, under root of 36 is 6, iska matlab, 6 square is 36.
11:10It is 36 only, okay.
11:13Let us check for 100 also, 100.
11:17If I divide by 2, it is 50, 50 divided by 2 is 25, 25 is divisible by 5, again like
11:27this.
11:28So that means under root of 100 we are finding, give me the answer quickly, we are finding
11:33under root of 100 which is under the root, 2, 2, 5 and 5.
11:41What did I say?
11:42You need to make the pair of the numbers, pair baniya.
11:46From each pair, 1 number will come out, isme se 2 out, isme se 5 out and this under root
11:54will be banished.
11:55So answer is 5 into 2, that is 10.
11:58It is verified because 10 square is what?
12:01It is 100.
12:02So what should be the under root of 100?
12:04That should be going back to this number, it is equal to 10.
12:09You can please practice this method prime factorization for the bigger numbers as well,
12:15okay.
12:16Now let's moving on to the next topic, square root by repeated subtraction.
12:23Sometimes the method will be listed, you have to use prime factorization or repeated subtraction
12:30or the long division.
12:31So it is very much necessary to first understand which method is being asked and then go ahead.
12:39Repeated subtraction, I'll do something, observe it carefully and then I'll explain.
12:45Repeated subtraction of consecutive odd numbers, thik hai, ye aap yaad rakhna, repeated subtraction
12:52of consecutive odd numbers only we take, 25 we are starting with which is the first odd
13:00number?
13:01It is 1, so 25 minus 1, it is 24, thik hai, continue with this number now, 24 minus, which
13:09is the next odd number after 1?
13:12It is 3, so 24 minus 3 is 21, continue with 21, which is the next odd number after 3?
13:20It is 5.
13:21So what is 21 minus 5?
13:23It is equal to 16, continue with 16, minus 7, 16 minus 7 is how much?
13:32It is 9, now continue with 9 minus the next odd number after 7 is 9, it is getting me
13:390.
13:40Once you get 0, you have to stop, thik hai, humne kya kiya?
13:45From the given number, we have subtracted the series of the consecutive odd numbers
13:50and we have picked the initial number as this the answer, thik hai, how many steps
13:55have we taken to complete this?
13:581, 2, 3, 4, 5, count the number of steps, 5 steps we have taken, so that means this
14:07number 25, it is the square of this 5 or we can say that under root of 25 is equal to
14:17the number of steps and how many steps we have taken, answer is 5, thik hai, you can
14:24do this for other numbers as well, just keep subtracting 1, 3, 5, 7, 9, 11 accordingly
14:31and count the number of steps, jitna bhi step hoga that is your under root.
14:36This is repeated subtraction method.
14:40Coming on to the one of the most important method that is by long division.
14:46The number here is 144, we will do for both the numbers, please mind it is a 3 digit number.
14:55Step number 1 has to be making the pair, thik hai, starting from the right hand side.
15:02From right hand side, this is the first pair, what alone is left out, let it be.
15:08So we are having 2 pairs of numbers starting from right hand side, first pair is 44, another
15:14single digit is 1.
15:16Now what you are going to do, you have to make a division table like this.
15:25Now after this what we are going to do, look for the number which multiplied by itself
15:32either should be equal to this number or lesser than this one, it is 1 only.
15:37So 1 into 1, it will give me 1, if I take 2, so 2 into 2 is 4 but it is greater than
15:441, so I have to start with 1, 1 multiplied by 1 is 1, difference of 1 and 1 is 0, I hope
15:53till here it is understood.
15:56Now this second pair, not 1 by 1, the whole pair will go down, so now my number is 44,
16:03okay.
16:04Take the double of the number at the top, you have to take the double of the number,
16:10if it was 2 here, you would have taken 4 here, it is 1, double of 1 is 2, I am taking
16:172 here.
16:18Now put a blank, I have to put a digit here such that if it is 2 blank, if I put 1 here,
16:26so 21 into 1, if I put 2 here, so 22 into 2, if I put 3 here, so 23 into 3, it has to
16:37be calculated like this.
16:39Now 21 into 1 is 21, it is lesser, 22 into 2, it is 44, am I getting the same number?
16:48So I will put 2 here, 2 at the top, so 22 into 2, I am getting 44 and the difference
16:57is 0.
16:58So this is your answer, under the root 144 is equal to 12, okay.
17:06Let's do one more question based on the same font set.
17:09I am taking a 4 digit number here, it is 3136.
17:16Step number 1, making the pair of 22 numbers starting from the right hand side, 2 numbers
17:21here and 2 numbers here.
17:24Now make a division table like this.
17:29After division table, what we are going to do, look for 2 similar numbers, if multiplied,
17:35you should get either this 31 or something lesser than this.
17:39Now look, if I take 4 square, that is 16, lesser, if I take 5 square, that is 25, if
17:47I take 6 square, that is 36, but 36 is greater than 31, I don't have to go till here.
17:54You have to start from 5, 5 into 5, that is 25, this much is understood.
18:02Now take the difference of 31 and 25, that is equal to 6.
18:08After taking the difference, this whole pair has to go down, you have got 636.
18:17What did I say, now take the double of this number.
18:20What is the double of 5, that is 10, double means into 2.
18:25Now the beginner number has to be 10, after 10, I have to put a digit here, okay, and
18:33this number has to be multiplied by that digit, if I put 1 after 10, then 101 into 1, if I
18:42put 2, 102 into 2, if I put, let's say 4, 104 into 4, okay, so we have to check for
18:51different combinations to get this number.
18:54Now if I check for 4, 4 into 4 is 16, and 4 into 1 is 4, it is very much lesser.
19:03Now let us understand one thing, how do we get 6 as a digit on one's place, how do we
19:11get 6 at one place, ya toh main 4 ka square leti hoon, wo 16 banta hai, dekho 6 at one
19:19position and if I take 6 square, then I get 36, one's position pe 6 tabhi aata hai, when
19:27I am taking the square of either 4 or 6, it can be 14, 16, it can be 24, 26, anything,
19:34but one's digit has to be 4 and 6, we have just tried with 4, I am trying with 6 now,
19:41106, yaha pe maine agar 6 liya, toh one more 6 has to go up, let us check, 106 into 6,
19:496 into 6 is 36, 6 into 0 is 0, and 3 as it is, 6 into 1 is 6, am I getting the same number,
19:59so 106 into 6, your answer is 636 and it is solved, your answer is 56, okay, so it comes
20:09with practice, do more questions based on this long division method. Now, we are starting
20:15exercise 6.1, many parts are there, so few few parts we will be picking, other parts
20:22you can do easily yourself, question number 1 says, what will be the unit digit of the
20:29square of the following number, unit digit means one's digit, so if I take first part,
20:37it is asking if we find 81 square, what will be one's digit, I have told you already, I
20:46have explained in the previous video, we need not to find the complete number, hum kya karenge,
20:5281 into 81, just multiply these two, 1 into 1 is 1, yaha pe puch bhi number ho hardly
21:00makes any difference, because we are told, we are asked to find just the one's digit
21:06or the unit digit, so answer is 1 here, have a look here, let's say for third part, number
21:14given is 799, it is asking, if I find the square of this number, what will be the one's
21:21digit, hum kya karenge, 799 into 799, 9 into 9 is 81, 1 will come here at 8 go up, ab jo
21:32bhi hum calculation karein, whatever is the number we are least bothered, we have got the
21:37one's digit, so your answer is 1 here, now we can do this corollarying also, 1, 2, 3, 4, what
21:45will be the square, one's digit of the square of this number, 4 into 4 is 16 and 16 ka one's
21:53digit kya hota hai, 1, 2, 3, 4 into 1, 2, 3, 4, 4 into 4 is 16, 1 will go as a carryover and
22:04we have got our one's digit, answer is 6 here, once more, if you take the last part, if you
22:12take the square of this number, so 5 into 5 is 25, hume 25 mila, what is the one's digit,
22:195 here, so your answer is 5, I hope first question is understood, let's move to the
22:25second question, the following numbers are obviously not perfect square, you have to give
22:31reason, if you remember in the previous video, the first introduction video, I have already
22:38told you that the numbers ending with 2, 3, 7, 8, they are never the perfect square, perfect
22:47square numbers only end with 0, 1, 4, 5, 6, 9 and 0 has to be repeated even number of times,
22:54so agar koi number 2, 3, 7, 8 par finish hota hai, it can never be a perfect square, so let's have
23:02a look at the numbers which are ending with 2, 3, 7, 8, have a look, one's digit 7 cannot be
23:10perfect square, one's digit 3 cannot be, one's digit 8 cannot be, odd number of zeros cannot be,
23:211, 3, 7, 8, okay, it is coming with 2 cannot be, odd number of zeros again it cannot be,
23:30so you can write the reason, perfect square numbers only end with 0, 1, 3, 4, 6, 9 and it
23:38never, 0, 1, 4, 5, 6, 9 and it never ends with odd number of zeros or 7, 3 and this one, 2, okay, so you can
23:49either, if you even get confused, you can also find out the square root but please remember that
23:54these numbers, the square numbers never end with these digits, have a look, 0, 1, 4, 5, 6, 9, answer is
24:02yes but if it is 2, 3, 7, 8, it is absolutely no, 2, 3, 7, 8, absolutely no, in this particular question, we were
24:11getting those digits only, 2, 3, 7, 8 or the odd number of zeros, so your answer is no here, question
24:18number 3, the square of which of the following will be an odd number, how can we determine if
24:26the answer will be odd or even? We have to just find the ones digit like we have done in question
24:33number 1, if one's digit is 1, 3, 5, 7, 9, your answer will be odd otherwise even and I hope you all know how to find the
24:45ones digit, so first question here, question number 3, first part, number is 4, 3, 1, if I find the square of this
24:54number, I just have to square of 1 and square of 1 is always 1, so whatever your answer is, one's digit will be 1 only and it will be an odd number, okay, this much is understood, now
25:09I read that even you can also do like this only, let me do this one, if it is 3, 5, 7, 9, what is the square of 9?
25:1981, two digit number, what is the ones place for? Answer will again be odd, okay, let's move to question number 4, you have to observe this pattern now and you have to fill in the numbers here, now you see when it is 1, 0, 1's square, 1 is repeated 2 times, so we have 1 till 2, 1, 0, 2, 0, 1, if I have 2 zeros here in between and 2 ones,
25:49we have 1 till 1, 2, we are just inserting how many zeros are there in between, now how many zeros are here? 4, so you have to insert 4 zeros here and 4 zeros here, okay, it is following the pattern, no zero, no zero, 1, 0, 1, 1, 0, 2 zeros, 2, 2, 0, now you can easily continue with this one, just count the number of zeros, 1,
26:16these many zeros, 2, these many zeros and 1, okay, question number 5, 11 square, 121, all these things are given to you, now how is it going, if 1, 1, 1 repeated 3 times, so we have 1 till 1, 2, 3 and the reverse order separated by zeros,
26:38now if I have number 1, 1, 1, 1, we are having 1 repeated 4 times, so 1, 2, 3 and 4 and then reverse order, okay, this is happening, 1, 2, 3, 3, 2, 1, 1 is repeated 4 times, so 1, 2, 3, 4 and the reverse order and you just need to insert zeros in between, okay and that's your answer.
27:05We are left with few more questions, like this one I have already discussed, 6, 7, 8 we are left with, 6, 7, 8, 9, just try to do these questions, if you will not be able to do these questions, I will make sure to discuss these questions in the next video, where we will be doing exercise 6.2 and 6.3, okay, practice these questions, it is just understanding based, I have already given you the formulas for doing the same,
27:35for example, 49, as the sum of 7 odd numbers, which series do we take, it is 1, 2, 3, 4, 5, 6 and 7, okay, 7 odd numbers are there and 7 square is 49, in the similar manner, if you continue this series till 11 odd numbers, you will get this answer,
27:59how many numbers lie between the square of these numbers, that means how many numbers between 12 square and 13 square, I told you double of the smaller number, so 2 times 12, answer is 24 and the same rule will be followed for these two, so please do not forget to complete the questions, repeat the video, re-watch it once again, if you are unable to understand or any doubts are there and please see the next session as well.
28:29For 6.2 and 6.3, please like, subscribe and follow my channel. Thank you so much for watching.