Is 1458 a perfect square if not find the smallest multiple of 1458 which is a perfect square also find the square root of new number?

Video Transcript

Find the smallest multiple of 62 d 72, which is a perfect square. Well, what i want to do is break this down into prime factors, so i want to start with 4 and see for 6272. Divided by 4 is going to be 1568 point. So i see that's going to be, then 4 will break down into 2 times 2. We have prime numbers i'm going to go to 601568 now and see with it. What is that divided by 2, but that's gonna be 784 and that's a prime number. Let'S take this divided by 2 and that's going to be 392, which is also a prime number. I wonder if that's divisible by 4. Let'S check, it is so that's 4 times 98, which 4 can break down into 2 times 2 point. So the diwan say 98 is 2 times 49, which is a prime number and then 49 is 7 times 7. So we have it broken down into prime numbers. Let'S list all this out, so 6272 is 123456722 times 2 times 2 times 2567 and then 27 is so to be a perfect square. We'Re going to have pairs of 2 point, so we have a pair of 2 pair of 2, a pair of 2 and a pair of sevens. Well, i see i'm missing a 2 to make this appear. So when i make this times 2, then i want to have another pair of 2 poison. So when i take 6272 times 2, that's going to end up being a perfect square. Specifically it's going to be 12544. What is that perfect square? Well, for each pair we're going to write down, we have a pair of 2 a pair of 2 a pair of 2 a pair of sevens and remember, we created a net of a pair of 2 s. So when we multiply all this together, we're going to end up with our perfect square or our square root of that perfect square, which is going to be 112.

1458 can be factorised as follows.

1458 = 2 × 3 × 3 × 3 × 3 × 3 × 3

Here, prime factor 2 does not have its pair. If 2 gets a pair, then the number will become a perfect square. Therefore, 1458 has to be multiplied with 2 to obtain a perfect square.

Therefore, 1458 × 2 = 2916 is a perfect square.

1458 × 2 = 2916 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

∴ `sqrt(2916)` = 2 x 3 x 3 x 3 = 54

Ex 6.3, 5 (v) - Chapter 6 Class 8 Squares and Square Roots

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Ex 6.3, 5 For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained. (v) 1458 Prime factorizing 1458 By prime factorization, 1458 = 2 × 3 × 3 × 3 × 3 × 3 × 3 Since 2 does not occur in pair we multiply by 2 to make it a pair So, our number becomes 1458 × 2 = 2 × 3 × 3 × 3 × 3 × 3 × 3 × 2 2916 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 Square root of 2916 ∴ √2916 = 2 × 3 × 3 × 3 = 6 × 9 = 54 ∴ The smallest whole number to be multiplied = 2 and square root of new number = 54

Solution:

(i) 252 = 2 x 2 x 3 x 3 x 7

Here, prime factor 7 has no pair. Therefore 252 must be multiplied by 7 to make it a perfect square.

\therefore252\times7=1764

And (i) \sqrt{1764}=2\times3\times7=42

(ii) 180 = 2 x 2 x 3 x 3 x 5

Here, prime factor 5 has no pair. Therefore 180 must be multiplied by 5 to make it a perfect square.

\therefore180\times5=900

And \sqrt{900}=2\times3\times5=30

(iii) 1008 = 2 x 2 x 2 x 2 x 3 x 3 x 7

Here, prime factor 7 has no pair. Therefore 1008 must be multiplied by 7 to make it a perfect square.

\therefore1008\times7=7056

And \sqrt{7056}=2\times2\times3\times7=84

(iv) 2028 = 2 x 2 x 3 x 13 x 13

Here, prime factor 3 has no pair. Therefore 2028 must be multiplied by 3 to make it a perfect square.

\therefore2028\times3=6084

And \sqrt{6084}=2\times2\times3\times3\times13\times13=78

(v) 1458 = 2 x 3 x 3 x 3 x 3 x 3 x 3

Here, prime factor 2 has no pair. Therefore 1458 must be multiplied by 2 to make it a perfect square.

\therefore1458\times2=2916

And \sqrt{2916}=2\times3\times3\times3=54

(vi) 768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

Here, prime factor 3 has no pair. Tehrefore 768 must be multiplied by 3 to make it a perfect square.

\therefore768\times3=2304

And \sqrt{2304}=2\times2\times2\times2\times3=48

Video transcript

Hello students, welcome to Lido Homework in this video we are going to solve this The following question that says we need to find the smallest whole number by which it should be multiplied rights which are the following number the smallest whole number by which of the following numbers should be multiplied so as to get a perfect square Okay, and once you find that number we will also have to find the square root so let's uh begin with the first one the first one says 252 okay so we know how to find the square root we are going to prime factorize it so let's take 252 and let's factorize this this goes into two two ones are two twos are two six are two six are two threes are this will go into three right three twos are three ones are three seven are and seven ones are so we've got all the factors so let's just write it down all the factors that are 2 into 2 into 3 into 3 into 7 okay after we write down the factors we need to group them so let's see if we are getting complete groups over here so we are getting yes a group of two, a group of three seven is a single number so now to complete this set i need to multiply one more seven so now i'm going to multiply one seven more that means i will need to multiply 7 with 252 so let's multiply 252 into 7 which gives me 1 7 6 4 okay so the number that you are multiplying with 252 is 7 that's the first part of the Question: The second part of the question says to also find the square root of the number obtained so the number obtained is 1764. so the square root of 1 1764 is nothing but square root of all these factors that we found over here so 2 into 2 into 3 into 3 into 7 and the 1 7 that we've multiplied now we know that we're going to group them and to find the square root we'll just take one number from each group so 2 into 3 into 7 which is equal to 42 so the square root of 1764 is 42 and the number multiplied with 252 is 7 to get to the nearest Perfect Square, let's do the second one the second one says 1 80 so let's factorize 180 this will go by 2. ninety-two uh fours are fives are ten okay carry forward there this will go by three three fifteen sir forty-five 3 5's are and 5 ones are okay so just let's write down the factors first 2 into 2 into 3 into 3 into 5 now let's group them and see which number is missing we have a group of 2 we have a Group 3 and a group of 5 is missing so I'll have to multiply by 5 to complete the group I'm going to multiply this by 5 that means i will have to multiply 180 by 5. multiplying 180 by 5 or you get 900 so 900 is the perfect square the second part of the question like mentioned says find out the square root of 900. so let's write that down in the square root bracket you're going to write all the factors okay 2 into 2 into 3 into 3 into 5 into 5 the next step would be taking one number from each group and when you do that you don't write the square root sign okay because now you've just taken out one number each and then you multiply 2 into 3 into 5 which is nothing but 30. so the square root of 900 is 30 and what number was supposed to be multiplied by 180 it was 5. i hope you're getting a hang of this feel free to pause the video solve the other sum by yourself comes back and check for yourself whether the answer is correct or not in this way it will just help you practice well okay uh let's move on to the third one that is one double zero eight so let's begin this clearly goes into two two fives are ten the zero is as it is and two fours are eight okay two twos are four two fives are ten two twos are four two ones are two twos are four carry one-two six are twelve two six are twelve two threes are six okay so 63 this goes by the next prime number and that is three three twos are six three ones are three three sevens are twenty-one seven ones are 7 okay so let's write down all the factors that we've got 2 into 2 into 2 into 2 into 3 into 3 into 7. now let's group them a group of 2 is there a group of 2 is There is a group of 3 that is there a group of 7 is missing so I'll have to multiply it by 7 to complete the group so if I'm going to add a 7 I'm going to multiply the original number one zero zero eight by seven which will give me seven zero five six so seven zero five six a perfect Square, let's find out the square root of seven zero five-six we already have the factors to put them under the square root two into two into two into two into three into three into 7 into 7 and we know the groups will take each number from each group so 2 into 2 into 3 into 7 the answer is 84 so the square root of 7056 is 84. Let's move on to the fourth one. that is 2 0 to 8. Okay, so let's factor this prime factorization this goes by 2. two are zeros are two are two fours are two fives are two zero and seven fourteen right this will go in by 1 6 and 9 169 now 169 is a perfect square of 13 right the more you practice the more you will be able to identify square roots directly okay otherwise um you will have to sit down and say okay five it doesn't go into five, it doesn't go into seven nine or eleven and then you come to thirteen okay Great, so this is the factor that we've received let's write down the factors 2 into 2 into 3 into 13 into 13 okay let's uh group them over here we can see There's a group of two There's a group of thirteen, a group of three is missing right so I'll have to multiply it by three so let's multiply two zero 2 8 by 3 into 3 the answer i'm going to get is 6 0 8 4 so 6 0 eight four is a perfect square for the second part of the question says to find out the square root of it is six zero eight four is equal to the square root of writing all the factors down under the square root sign, I'll write it like this: and then we also added a 3 okay you get this and then we'll select each one from each group so 2 into 13 into 3 you can also write it as 2 into 3 into 13 and then the answer for this is seventy eight okay so square root of six zero eight four is seventy eight let's move on to the fifth one one four five eight okay so let's solve this one four five eight clearly this goes by two two sevens are two twos are two nines are eighteen seven twenty nine let's try this with three going by three two four three right three eights are twenty four three ones are 81 again this will go in by three three twos are six three nines are no three sevens are 21 right yeah and then 3 nines are twenty seven three threes are nine three of them are three okay i'll just write this properly so this is 27 okay yeah so now we have the factors Now let's write down the factors 2 into 3 into 3 into 3 into 3 into 3 into 3 Okay, let's group them up I see a group of 3 I see a group of 3 i see a group of 3 a group of 2 is missing so I'll have to multiply it by 2 to complete the group of 2 right so since I'm multiplying it with 2 I will multiply 1 4 5 8 into 2 which I'll get as 2 9 1 6 so 2 9 1 6 is a perfect The second square part of this question asks us to find the square root, so 2 9 1 6 is equal to the square root of 2 9 1 6 is equal to square root of write down all the factors that we have 3 into 3 and 3 into 3 okay let's group them and take each number from each group 2 into 3 into 3 into 3 and the answer for this would be 54 okay so the square root of 2916 which is a perfect square that we got by multiplying 2 to 1 4 5 8. the square root of 2 9 1 6 is 54. let's go to the last one here and that is 768 this will go in by 2 384 to 192 again it will go in by two,, nines are eighteen two six are twelve two fours are eight two eights are sixteen two twos are four two fours are eight twenty four two ones are two twos are four two six are twelve two threes are six and we have three ones okay let's write down all the factors so we have 2 into 2 into 2 into 2 into 2 into 2 into 2 into 2. let's check whether we've written down all the numbers uh you might tend to miss out numbers so another thing you can do is while copying down uh the numbers you can just go with your pencil just marked the pairs over here so you know you're Copying down the right um number of twos okay so we have four pair so I should have four pairs are written down here so this one these two are three and 4 great and then I have a single 3 so now the number that needs to be multiplied is 3 because we need to complete this pair of 3 right so let's multiply 7 68 into 3 which gives me 2 3 0 4 so 2 3 0 4 is a perfect square let's find out the square root of 2 3 0 4 which is nothing but square root of all the factors we'll write under the square root sign 2 into 2 two into a two-second group the third group of two into two fourth groups of two into two and one group of three into three okay Now let's take a single number from each group so 2 into 2 into 2 into 2 into 3 when you multiply this You'll get the answer as 40 8. so the square root of 2 3 0 4 is 48 that's all in this video, please feel free to like to share and comment I'll see you in the next video Thank you.

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