Square Root of 1504

The square root is the inverse of the square of the number. 1504 is not a perfect square. The square root of 1504 is expressed in both radical and exponential form. In the radical form, it is expressed as √1504, whereas (1504)^(1/2) in the exponential form. √1504 ≈ 38.77886, 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 1504 is broken down into its prime factors.

Step 1: Finding the prime factors of 1504. Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 47: 2^5 × 47.

Step 2: Now we found out the prime factors of 1504. The second step is to make pairs of those prime factors. Since 1504 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1504 using prime factorization to find an exact square root is impossible.

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 1504, we need to group it as 04 and 15.

Step 2: Now we need to find n whose square is less than or equal to 15. We can say n is '3' because 3 × 3 = 9 is less than 15. Now the quotient is 3, and after subtracting 9 from 15, the remainder is 6.

Step 3: Bring down 04 to make it 604 as the new dividend. Add the old divisor with the same number 3 + 3 = 6 which will be our new divisor.

Step 4: Find a digit x such that 6x × x is less than or equal to 604. Let’s choose x as 9, now 69 × 9 = 621, which is too large, so we try x as 8, giving us 68 × 8 = 544.

Step 5: Subtract 544 from 604, the difference is 60, and the quotient is 38.

Step 6: 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 6000.

Step 7: Find the new divisor which is 769 because 769 × 7 = 5383.

Step 8: Subtract 5383 from 6000, we get the result 617.

Step 9: Now the quotient is 38.7.

Step 10: Continue doing these steps until we get the desired decimal places.

So, the square root of √1504 ≈ 38.77886.

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 1504 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 1504.

The smallest perfect square less than 1504 is 1444 (38^2) and the largest perfect square greater than 1504 is 1521 (39^2). √1504 falls between 38 and 39.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Using the formula (1504 - 1444) ÷ (1521 - 1444) = 60 ÷ 77 ≈ 0.779. Therefore, 38 + 0.779 = 38.779, so the square root of 1504 is approximately 38.779.

• Square root: A square root is the inverse of a square. For 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 a principal square root.

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

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.