Question

Find the square root of the following numbers using the prime factorization method
A) 4225
B) 10.24
C) 0.023104
D) \[22\dfrac{9}{{16}}\]

Answer 1

Hint:As we are mentioned to find the factors using the prime factorisation method then our main motive is to break the numbers into all the prime factors and then those numbers who have 2 same prime factors will become one. Like for example \[4 = 2 \times 2\] then the square root of 4 is 2.

Complete step by step answer:
A) The number given to us is 4225
Now 4225 can be first divided by 5 and we will get 854 dividing it again by 5 we will get 169, now 169 is divisible by 13 only so after dividing we are getting 13, which means that \[4225 = 5 \times 5 \times 13 \times 13\] and hence after squarooting we will have \[5 \times 13 = 65\]
B) The number given to us is 10.24
Which can also be written as \[\dfrac{{1024}}{{100}}\] Now we know that \[10 \times 10 = 100\] and for 1024 just keep on dividing the number by 2 it's a very special number s we get the value of \[1024 = {2^{10}}\] which means that we are multiplying 2, ten times. So if we try to find the value of \[\sqrt {{2^{10}}} \] we will get it as \[{{2^5}}\] which is 32. So now the square root of \[\sqrt {\dfrac{{1024}}{{100}}} = \dfrac{{32}}{{10}}\] so that means 3.2 is the correct answer.
C) Given number is 0.023104
Just like the last number it can also be written as \[\dfrac{{23104}}{{1000000}}\] So square root of \[1000000 = 1000\] and \[23104 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 19 \times 19\] Which means that \[\sqrt {23104} = 2 \times 2 \times 2 \times 19 = 152\]
D) We are given the number \[22\dfrac{9}{{16}}\] which is in mixed dfraction so converting it into improper dfraction we will have \[22\dfrac{9}{{16}} = \dfrac{{22 \times 16 + 9}}{{16}} = \dfrac{{361}}{{16}}\] Now we know that 16 is \[{4 \times 4}\] so square root will be 4 only and that for 361 it will be \[19 \times 19\] so its square root will be 19 and hence we are getting the number as \[\dfrac{{19}}{4} = 4\dfrac{3}{4}\]

Note:
 Be conscious about the dividing and always divide with the lowest number possible because that will eventually help to get the prime factors clearly out of the number and then we can easily find out the square roots.