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Square root of 128
The square root of 128 is a number which when multiplied by itself, results in the number 128. The value of a square root can be a decimal or a whole number, or an integer. We will now look at how to calculate the square root of 128 and see a few interesting problems.128 is also represented as 2^{7}. In this lesson, we will calculate the square root of 128 by long division method.
 Square Root of 128: √128 = 11.313
 Square of 128: 128² = 16,384
What Is the Square Root of 128?
The square root of a number is the number that when multiplied to itself gives the original number as the product. i.e. a × a = n and it is a^{2 }= n. Finding square root is the inverse process of squaring a number. Therefore, a = √n
 Radical form: 128 = 2^{7 }= 2 × 2 × 2 × 2 × 2 × 2 × 2 ⇒ √128 = √(2 × 2 × 2 × 2 × 2 × 2 × 2 ) = 2 × 2 × 2 × √2 = 8 √2
 Exponential form: 128^{½}
 Approximate value : 11.313
Is Square Root of 128 Rational or Irrational?
A rational number is defined as a number that can be expressed in the form of a quotient or division of two integers, i.e. p/q, where q is not equal to 0. √128 = 11.31370849898476 nonterminating decimal and hence it is proved that it is an irrational number.
How to Find the Square Root of 128?
The square root of 128 can be calculated using different methods such as prime factorization or the long division method or the approximation method.
Square Root of 128 by Approximation Method
 128 lies between the two perfect squares 121 and 144. 121 < 128 < 144
 √121 < √128 < √144
 11 < 128 < 12
 Now divide 128 by 11 or 12. Let us divide by 11. 128 ÷ 11 = 11.63
 Find the average of the quotient so obtained with 11. (11.63 + 11) ÷ 2 = 22.63 ÷ 2 = 11.315
 √128 ≈ 11.315
Square Root of 128 by Long Division Method
 Step 1: Write 128 as 128000000. Select the numbers in pairs from the right. We have 1 alone in the left.
 Divide 1 by 1 and get the remainder 0. Quotient is 1. Bring down the next pair, 28. 28 is the new dividend.
 Step 2: Double the quotient. Have 20 in the new divisor's place.
 Find a number which when added to 20 and the sum multiplied with the same number gives 28 or less than that. 1 added to 20 is 21 and 21 × 1 = 21.
 Subtract 21 from 28 and get the remainder as 7. Bring down the next pair of zeros and have 700 as the new dividend.
 Step 3: The quotient now is 11. Double it. We get 22. Have 220 at the new divisor's place.
 Determine a number which when added to 220 and the sum multiplied with that number gives the product 700 or less than that. Clearly, 223 × 3 = 669.
 Subtract 669 from 700. Obtain the remainder 31.
 Now repeat the process until you approximate the quotient to 3 decimal places.
Explore Square roots using illustrations and interactive examples:
Tips And Tricks
 128 lies between the two perfect squares √121 and √144 and hence √128 lies between 11 and 12.
 Use the average method to approximate the evaluate √128.
Important Notes
 The √128 in the simplest radical form is 8√2 and the exponential form 128 ^{½}
 √128 = 11.313
 √128 is irrational.
Square Root of 128 Solved Examples

Example 1: Evaluate: √128  √32
Solution:
√128 = √(64 × 2) = 8 √2
√32 = √(16 × 2) = 4 √2
√128  √32 = (8 + 4 ) √2
= 12 √2
Therefore, √128  √32 = 12√2 
Example 2 : The area of a square tile is 12800 sq inches. How long is one side of this tile to the nearest tenth of an inch?
Solution:
Area = side × side sq inches
12800 = side × side
Side^{2} = 12800
Taking the square root on both the sides, we get
(side ^{2}) ^{½} = (12800)^{½}
side = √128 × √100
Approximating √128 to the nearest tenth, we get √128 = 11.3
side = 11. 3 × 10 = 113 inches.
The side of the square tile = 113 inches.
FAQs on Square Root of 128
What is the square root of 128?
The square root of 128 is 11.313.
Why is the square root of 128 simplified?
The square root of 128 simplified is 8 √2.
Is 128 rational or irational?
128 is rational, as it is expressed as 128/1 in the rational form. However square root of 128 is 11.3 and it is irrational.
Between which two whole numbers does the square root of 128 lie?
The square root of 128 lies between 11 and 12.
How do we find the square root of 128?
We use the approximation method or the long division method to get the square root of 128.
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