Square Root of 134

The square root is the inverse of the square of the number. 134 is not a perfect square. The square root of 134 is expressed in both radical and exponential form. In the radical form, it is expressed as √134, whereas (134)^(1/2) is the exponential form. √134 ≈ 11.57584, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 134, we use the long-division method and approximation method. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now, let us look at how 134 is broken down into its prime factors.

Step 1: Finding the prime factors of 134 Breaking it down, we get 2 x 67: 2¹ x 67¹

Step 2: Now we found out the prime factors of 134. Since 134 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √134 using prime factorization is not possible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 134, we need to group it as 34 and 1.

Step 2: Now we need to find n whose square is less than or equal to 1. We can say n is ‘1’ because 1 x 1 is less than or equal to 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 34, which is the new dividend. Add the old divisor with the same number, 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 34. Let us consider n as 1, then 21 x 1 = 21.

Step 6: Subtract 34 from 21; the difference is 13, and the quotient is 11.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1300.

Step 8: Now we need to find the new divisor that is 115 because 2315 x 5 = 11575, which is closest to 13000.

Step 9: Subtracting 11575 from 13000, we get the result 1425.

Step 10: Now the quotient is 11.57.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values continue till the remainder is zero.

So the square root of √134 is approximately 11.575.

The approximation method is another method for finding the square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 134 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √134. The smallest perfect square less than 134 is 121 and the largest perfect square greater than 134 is 144. √134 falls somewhere between 11 and 12.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (134 - 121) ÷ (144 - 121) = 0.565 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 11 + 0.565 = 11.565.

Thus, the square root of 134 is approximately 11.565.

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse operation is the square root, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that is used in most applications. Therefore, it is also known as the principal square root.
   
• Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 134 is 2 x 67.
   
• Long division method: A method used to find the square root of non-perfect square numbers by performing a series of division steps.