Square Root of 1515

The square root is the inverse of the square of the number. 1515 is not a perfect square. The square root of 1515 is expressed in both radical and exponential form. In the radical form, it is expressed as √1515, whereas (1515)^(1/2) in the exponential form. √1515 ≈ 38.923, 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 1515 is broken down into its prime factors.

Step 1: Finding the prime factors of 1515 Breaking it down, we get 3 × 5 × 101: 3¹ × 5¹ × 101¹

Step 2: Now we found out the prime factors of 1515. The second step is to make pairs of those prime factors. Since 1515 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1515 using prime factorization is not possible in terms of a perfect square.

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 1515, we need to group it as 15 and 15.

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

Step 3: Now let us bring down the next group, which is 15, making the new dividend 615. Add the old divisor with the same number 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor will be 60n, where n is our next digit in the quotient.

Step 5: Find n such that 60n × n ≤ 615. Let us consider n as 1. Now 601 × 1 = 601.

Step 6: Subtract 601 from 615, the remainder is 14, and update the quotient to 31.

Step 7: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend, making it 1400.

Step 8: Now find the new divisor, which is 620, because 6202 × 2 = 1240.

Step 9: Subtracting 1240 from 1400 gives 160.

Step 10: The quotient is now 38.92.

Step 11: Continue these steps until you get the desired precision.

So the square root of √1515 ≈ 38.923.

The approximation method is another method for finding square roots; it is an easy method to approximate the square root of a given number. Now let us learn how to approximate the square root of 1515 using this method.

Step 1: Find the closest perfect squares surrounding √1515.

The smallest perfect square less than 1515 is 1444 (38²), and the largest perfect square greater than 1515 is 1521 (39²). √1515 falls between 38 and 39.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).

Using the formula, (1515 - 1444) / (1521 - 1444) = 71 / 77 ≈ 0.922 Adding this decimal to the lesser whole number: 38 + 0.922 = 38.922.

Therefore, the square root of 1515 is approximately 38.923.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: 4² = 16 and the inverse of the square is the square root, that is, √16 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction, meaning it 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: Although a number has both positive and negative square roots, the positive square root is often used in practical applications. It is known as the principal square root.

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

• Long division method: A technique used to find the square root of a non-perfect square by dividing the number into pairs of digits from right to left and performing a series of divisions to approximate the square root.