Square Root of 147

The square root of 147 is ±12.124. Finding the square root of a number is the inverse process of finding the perfect square. The square root of 147 is written as √147. 

The different ways to find the square root of a number are prime factorization, long division  and approximation/estimation method
 

The prime factorization of 147 breaks 147 into its prime numbers. 

The numbers 3 and 7 are the prime numbers 

Prime factorization of 147 is 3 × 7²

Since 7 is repeating, we can pair 7. 

Therefore, √146 is expressed as 7√3, the simplest radical form

The long division method finds the square root of non-perfect squares.

Step 1: Write down the number 147

Step 2: Number 147 is a three-digit number, so pair them as (1), (47)

Step 3: Find the largest that is closest to the first pair (1), which is 1²

Step 4: Write down 1 as the quotient, which will be the first digit of the square root.

Step 5: Subtracting 1²  from 1 will leave zero as the remainder. Now bring down the second pair (47) and place it beside 0.

Step 6: Now double the quotient you have, that is multiply the quotient 1 with 2 and the result will be 2

Step 7: Choose a number such that it can be placed after 2. The two-digit number created should be less than the second pair (47). Here, we place number 1 after 2, because the number formed is less than 47.

Step 8: Subtract 22 from 47  → 47-22 = 3. Now add a decimal point after the new quotient and adding two zeros will make it 300

Step 9: Apply step 7 over here and continue the process until you reach 0.

Step 10: We can write  √147 as 12.124

The approximation method finds the estimated square root of non-perfect squares.

Step 1: Identify the closest perfect square to 147. Numbers 144 and 169 are the closest perfect square to 147.

Step 2: We know that  √144 = 12 and  √169 = 13. Thus, we can say that  √147 lies between  12 and 13.

Step 3: Check if  √147 is closer to 12 or 13. Let us take 12.5 and 13. Since (12.5)² is 156.25 and (13)² is 169,  √147 lies between them.

Step 4: We can keep changing the values of 12.5 to 12. 6 and iterate the same process without changing 13 as the closest perfect square root.

The result of  √147 will be 12.124
 

• Perfect Square: Product obtained when the same number gets multiplied twice

• Approximate Value: Value closer to the exact number

• Prime Factorization: Breaking down the number into its prime factors.