Square Root of 1575

The square root is the inverse of the square of a number. 1575 is not a perfect square. The square root of 1575 is expressed in both radical and exponential form. In the radical form, it is expressed as √1575, whereas (1575)^(1/2) in the exponential form. √1575 ≈ 39.6863, 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 1575 is broken down into its prime factors:

Step 1: Finding the prime factors of 1575 Breaking it down, we get 3 x 3 x 5 x 5 x 7: 3² x 5² x 7.

Step 2: Now we found out the prime factors of 1575. The second step is to make pairs of those prime factors. Since 1575 is not a perfect square, calculating the square root using prime factorization involves using the pairs 3² and 5².

Therefore, the square root of 1575 is 3 x 5 x √7 ≈ 39.6863.

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 1575, we need to group it as 75 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 x 3 = 9 is lesser than 15. Now the quotient is 3, after subtracting 9 from 15, the remainder is 6.

Step 3: Now let us bring down 75, which is the new dividend. Double the quotient and use it as the first digit of our trial divisor. So we have 6_.

Step 4: Find n such that 6n x n is less than or equal to 675. Trying n as 9, we have 69 x 9 = 621.

Step 5: Subtract 621 from 675, the difference is 54, and the quotient is 39.

Step 6: Since the remainder is less than the divisor, add a decimal point and bring down two zeros.

Step 7: Continue this process until the desired precision is achieved.

The result is 39.6863.

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 1575 using the approximation method:

Step 1: Now we have to find the closest perfect square of √1575.

The smallest perfect square close to 1575 is 1521 (39²) and the largest is 1600 (40²).

√1575 falls between 39 and 40.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (1575 - 1521) / (1600 - 1521) = 54 / 79 ≈ 0.68.

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 39 + 0.68 ≈ 39.68, so the square root of 1575 is approximately 39.68.

• 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 the reason it is also known as the principal square root.
   
• Prime factorization: Prime factorization involves expressing a number as a product of its prime numbers. For example, the prime factorization of 1575 is 3² x 5² x 7.
   
• Long division method: The long division method is a technique to find the square root of a number by dividing it into groups of digits and performing a series of steps to achieve the desired precision.