Square Root of 1875

The square root is the inverse of the square of the number. 1875 is not a perfect square. The square root of 1875 is expressed in both radical and exponential form. In the radical form, it is expressed as √1875, whereas (1875)^(1/2) in the exponential form. √1875 ≈ 43.30127, 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 1875 is broken down into its prime factors:

Step 1: Finding the prime factors of 1875 Breaking it down, we get 3 × 5 × 5 × 5 × 5: 3^1 × 5^4

Step 2: Now we found out the prime factors of 1875. The second step is to make pairs of those prime factors. Since 1875 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely to simplify the square root.

Therefore, calculating √1875 using prime factorization gives us an approximate value.

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 1875, we need to group it as 75 and 18.

Step 2: Now we need to find n whose square is less than or equal to 18. We can say n as ‘4’ because 4 × 4 = 16, which is less than 18. Now the quotient is 4, and after subtracting 16 from 18, the remainder is 2.

Step 3: Now let us bring down 75, which is the new dividend. Add the old divisor with the same number 4 + 4 to get 8, which will be part of our new divisor.

Step 4: The new divisor will be 8n. We need to find the value of n such that 8n × n is less than or equal to 275 (the new dividend).

Step 5: By trial, n is 3 because 83 × 3 = 249.

Step 6: Subtract 249 from 275, the difference is 26, and the quotient is 43.

Step 7: Since the remainder is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2600.

Step 8: Continue performing division until you reach an estimated value. After a few more steps, the quotient will be approximately 43.301.

So the square root of √1875 is approximately 43.301.

The approximation method is another method for finding square roots, and 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 1875 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √1875. The smallest perfect square less than 1875 is 1764 (42^2), and the largest perfect square greater than 1875 is 1936 (44^2). √1875 falls somewhere between 42 and 44.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula, (1875 - 1764) / (1936 - 1764) = 111 / 172 ≈ 0.645 The approximate square root is the lower bound plus the decimal: 42 + 0.645 ≈ 42.645, so the square root of 1875 is approximately 43.301.

• 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, which 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, the principal square root is the non-negative root commonly used in real-world applications.

• Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 1875 is 3 × 5^4.

• 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.