Square Root of 375

The square root is the inverse of the square of the number. 375 is not a perfect square. The square root of 375 is expressed in both radical and exponential form. In the radical form, it is expressed as √375, whereas (375)^(1/2) in the exponential form. √375 ≈ 19.3649, 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 375 is broken down into its prime factors.

Step 1: Finding the prime factors of 375 Breaking it down, we get 3 × 5 × 5 × 5: 3^1 × 5^3

Step 2: Now we found out the prime factors of 375. The second step is to make pairs of those prime factors. Since 375 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating √375 using prime factorization 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 375, we need to group it as 75 and 3.

Step 2: Now we need to find n whose square is 3. We can say n as ‘1’ because 1 × 1 is lesser than or equal to 3. Now the quotient is 1; after subtracting 1 from 3, 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 1 + 1 we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 275; let us consider n as 9; now 29 × 9 = 261

Step 6: Subtract 261 from 275; the difference is 14, and the quotient is 19.

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 1400.

Step 8: Now we need to find the new divisor that is 389 because 389 × 3 = 1167

Step 9: Subtracting 1167 from 1400, we get the result 233.

Step 10: Now the quotient is 19.3

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √375 is approximately 19.36.

The approximation method is another method for finding the 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 375 using the approximation method.

Step 1: Now we have to find the closest perfect square of √375.

The smallest perfect square of 375 is 361 and the largest perfect square of 375 is 400.

√375 falls somewhere between 19 and 20.

Step 2: Now we need to apply the formula that is

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

Going by the formula (375 - 361) ÷ (400 - 361) = 0.36

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 19 + 0.36 = 19.36, so the square root of 375 is approximately 19.36.

• 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, 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: The process of expressing a number as a 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.