Question

Find the square root of the following number: 1024

Answer 1

Hint: Let us find the prime factors by the given number 1024 by dividing the number by the smallest prime factor till we get a number which is not divisible.

Complete step-by-step answer:
As we know to find the square root of the number 1024. First, we had to split this number into its different prime factors.
And after that we had to make pairs of two similar factors and multiply them to get the required square root of the number.
Now, for finding prime factors we had to divide the number by the smallest prime number by which it is divisible.
So, for finding prime factors.
Now we have to divide the number by 2 till we get a number which is not divisible by 2.
Dividing the number by 2. We get \[\dfrac{{1024}}{2} = 512\]
Now again dividing the above result by 2. We get \[\dfrac{{512}}{2} = 256\]
Now again dividing the above result by 2. We get \[\dfrac{{256}}{2} = 128\]
Now again dividing the above result by 2. We get \[\dfrac{{128}}{2} = 64\]
Now again dividing the above result by 2. We get \[\dfrac{{64}}{2} = 32\]
Now again dividing the above result by 2. We get \[\dfrac{{32}}{2} = 16\]
Now again dividing the above result by 2. We get \[\dfrac{{16}}{2} = 8\]
Now again dividing the above result by 2. We get \[\dfrac{8}{2} = 4\]
Now again dividing the above result by 2. We get \[\dfrac{4}{2} = 2\]
Now again dividing the above result by 2. We get \[\dfrac{2}{2} = 1\]
So, the prime factors of 1024 are \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
Now we had to find the square root of the given number.
So, \[\sqrt {1024} = \sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} = \sqrt {{{\left( 2 \right)}^{10}}} = {\left( 2 \right)^5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\]
Hence, the square root of 1024 will be 32.

Note: Whenever we come up with this type of problem then first, we have to find all the prime factors of the given number by dividing the number by the smallest prime number by which it is divisible, and then we can also directly find the square root of the number by directly replacing two pairs of the same factors with one factor and then multiplying all. Like if factors of any number is \[{\text{a}} \times {\text{a}} \times {\text{a}} \times {\text{a}} \times {\text{b}} \times {\text{b}} \times {\text{b}} \times {\text{b}}\] then its square root will be \[{\text{a}} \times {\text{a}} \times {\text{b}} \times {\text{b}}\] because a and b both occurs four times in the factors.