Question

What is the square root of \[7744\]?

Answer 1

Hint:We need to find the square root of \[7744\]. We will use the prime factorisation method to find the square root of \[7744\]. For that, we first need to dissolve the given number into prime factors and write it as a product of prime numbers. After that, we need to make a pair of similar factors. Now, to find the square root of the given number, we need to extract one out of every pair and then the product of these numbers give us the square root of the given number.

Complete step by step answer:
We need to find the square root of \[7744\].
Firstly, let us write \[7744\] as a product of prime numbers.
Now, \[7744\] can be written as
\[ \Rightarrow 7744 = 2 \times 3872\]
Further, we can decompose \[3872\] and write it as \[3872 = 2 \times 1936\]. Using this, we get
\[ \Rightarrow 7744 = 2 \times 2 \times 1936\]

Now writing \[1936 = 2 \times 968\], we get
\[ \Rightarrow 7744 = 2 \times 2 \times 2 \times 968\]
Further writing \[968 = 2 \times 484\], we get
\[ \Rightarrow 7744 = 2 \times 2 \times 2 \times 2 \times 484\]
Now, writing \[484 = 2 \times 242\], we get
\[ \Rightarrow 7744 = 2 \times 2 \times 2 \times 2 \times 2 \times 242\]
We know, \[242\] can be written as \[242 = 2 \times 121\]. So, using this, we get
\[ \Rightarrow 7744 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 121\]
Now, we know, \[121 = 11 \times 11\]. So, writing \[121\] as \[11 \times 11\], we get
\[ \Rightarrow 7744 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 \times 11\]

Now, pairing the numbers, we get
\[ \Rightarrow 7744 = \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {11 \times 11} \right)\]
Now, taking out one out of every pair and multiplying those numbers gives us the square root of \[7744\].Hence, we get, square root of \[7744\]
\[ \Rightarrow \sqrt {7744} = \left( 2 \right) \times \left( 2 \right) \times \left( 2 \right) \times \left( {11} \right)\], where \[\sqrt {7744} \] is the square root of \[7744\].
Multiplying the numbers, we get
\[ \therefore \sqrt {7744} = 88\]

Hence, the square root of \[7744\] is \[88\].

Note:Here, we solved this by the Prime factorisation method but we could have solved this question using the long division method. Steps for Long Division Method are as follows:
Step 1: Starting from the right, start making a pair of two digits for the given number. Note that, we can be left with a single digit in the left most corner. For example, when we are given a three digit number, we will have one pair and a single digit in the left most corner.
Step 2: Now, we need to think of the greatest number such that its square is less than or equal to the first pair or the leftmost digit taking this number as the quotient and the divisor.
Step 3: Subtracting the product of the divisor and the quotient from the first pair or the first digit, we bring down the next pair with the remainder and now this digit becomes our next dividend.
Step 4: For the next divisor, we will first double the quotient and then think of a number such that when this number is added to ten times the double of quotient and then multiplied with itself gives less than or equal to the new dividend. Also, this number becomes the next digit of the quotient.
Step 5: Repeat the above steps for the next pair and proceeding further we get our quotient as the square root of the given number.