Square Root of 1500

The square root is the inverse of the square of the number. 1500 is not a perfect square. The square root of 1500 is expressed in both radical and exponential form. In the radical form, it is expressed as √1500, whereas (1500)^(1/2) in the exponential form. √1500 ≈ 38.72983, 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 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 1500 is broken down into its prime factors.

Step 1: Finding the prime factors of 1500 Breaking it down, we get 2 x 2 x 3 x 5 x 5 x 5: 2^2 x 3^1 x 5^3

Step 2: Now we found out the prime factors of 1500. The second step is to make pairs of those prime factors. Since 1500 is not a perfect square, the digits of the number can’t be grouped in pairs perfectly.

Therefore, calculating 1500 using prime factorization directly is not feasible.

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 1500, we need to group it as 00 and 15.

Step 2: Now we need to find n whose square is 15. We can say n as ‘3’ because 3 x 3 is lesser than or equal to 15. Now the quotient is 3, after subtracting 9 from 15, the remainder is 6.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number, 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 600. Let us consider n as 9, now 69 x 9 = 621.

Step 6: Subtract 621 from 600, the difference is 21, and the quotient is 38.

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 2100.

Step 8: Now we need to find the new divisor that is 387 because 387 x 5 = 1935.

Step 9: Subtracting 1935 from 2100, we get the result 165.

Step 10: Now the quotient is 38.7.

Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √1500 is approximately 38.73.

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 1500 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1500.

The smallest perfect square less than 1500 is 1444 (which is 38^2), and the next perfect square is 1521 (which is 39^2). √1500 falls somewhere between 38 and 39.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (1500 - 1444) ÷ (1521 - 1444) = 56 / 77 ≈ 0.727.

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 38 + 0.727 ≈ 38.73.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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, 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 the reason it is also known as the principal square root.

• Prime factorization: Prime factorization is the process of expressing a number as a product of its prime factors.

• Long division method: A method used to divide numbers to find a square root by approximating the quotient step by step.