Square Root of 128

The square root of 128 is ±11.3137. Finding the square root of a number is the inverse process of finding the perfect square. The square root of 128 is written as √128. 
 

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

The prime factorization of 128 breaks 128 into its prime numbers. 

The number 2 is the only prime number

Prime factorization of 128 is 2⁷

Since 2 is repeating, we can pair them

Therefore, √128 is expressed as 2³ x √2 →8√2, the simplest radical form.
 

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

Step 1: Write down the number 128

Step 2: Number 128 is a three-digit number, so pair them as (1), (28)

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 (28) and place it beside 0.

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

Step 7: Choose a number such that it can be placed after 2 in the divisor. The two-digit number created should be less than the second pair (28). Here, we add 1 after 2, because the number formed will be less than 28.

Step 8: Subtract 21 from 28  → 28-21 =7. Now add a decimal point after the new quotient and adding two zeros will make it 700

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

Step 10: We can write  √128 as 11.313.

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

Step 1: Identify the closest perfect square to 128. Numbers 121 and 144 are the closest perfect square to 128.

Step 2: We know that  √121 = 11 and  √144 = 12. Thus, we can say that  √128 lies between  11 and 12.

Step 3: Check if  √128 is closer to 11 or 12. Let us take 11.5 and 12. Since (11.5)² is 132.25 and (12)² is 144,  √119 lies between them.

Step 4: We can keep changing the values of 11.5 to 11. 6 and iterate the same process without changing 12 as the closest perfect square root.

The result of  √128 will be 11.313
 

• 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.