Square Root of 145

The square root is the inverse of the square of the number. 145 is not a perfect square. The square root of 145 is expressed in both radical and exponential form. In the radical form, it is expressed as √145, whereas (145)^(1/2) in the exponential form. √145 ≈ 12.0416, 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, the prime factorization method is not used for non-perfect square numbers, where the long division method and approximation method are used. 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 145 is broken down into its prime factors.

Step 1: Finding the prime factors of 145 Breaking it down, we get 5 × 29.

Step 2: Now we found out the prime factors of 145. The second step is to make pairs of those prime factors. Since 145 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 145 using prime factorization alone will not give an exact square root.

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 145, we need to group it as 45 and 1.

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1×1 is equal to 1. Now the quotient is 1; after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 45, which is the new dividend. Add the old divisor with the same number 1 + 1, which we get 2 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 ≤ 45. Let us consider n as 2, now 22 = 4 which gives 24 when multiplied by 2, giving 44.

Step 6: Subtract 45 from 44, the difference is 1, and the quotient is 12.

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 100.

Step 8: Now we need to find the new divisor that is 24, because 244 × 4 = 976.

Step 9: Subtracting 976 from 1000, we get the result 24.

Step 10: Now the quotient is 12.04.

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

So the square root of √145 is approximately 12.04.

The approximation method is another method for finding 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 145 using the approximation method.

Step 1: Now we have to find the closest perfect square of √145. The smallest perfect square less than 145 is 144, and the largest perfect square greater than 145 is 169. √145 falls somewhere between 12 and 13.

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 (145 - 144) ÷ (169 - 144) = 0.04. 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 12 + 0.04 = 12.04.

So the square root of 145 is approximately 12.04.

• Square root: A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root, that is √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 has more prominence due to its uses in the real world. That is why it is known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 145 is 5 × 29.
   
• Long division method: A method used to find the square root of non-perfect squares by systematically dividing the number into parts to approximate the root.