Square Root of 3375

The square root is the inverse of the square of the number. 3375 is not a perfect square. The square root of 3375 can be expressed in both radical and exponential forms. In radical form, it is expressed as √3375, whereas in exponential form it is expressed as (3375)^(1/2). √3375 = 58.09475, 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 3375 is broken down into its prime factors.

Step 1: Finding the prime factors of 3375 Breaking it down, we get 3 x 3 x 3 x 3 x 5 x 5 x 5: 3^4 x 5^3

Step 2: Now we found out the prime factors of 3375. The second step is to make pairs of those prime factors. Since 3375 is not a perfect square, the digits of the number can’t be grouped in equal pairs. Therefore, calculating 3375 using prime factorization directly is complex.

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 3375, we need to group it as 75 and 33.

Step 2: Now we need to find a number whose square is less than or equal to 33. We can say this number is '5' because 5 x 5 = 25, which is less than 33. After subtracting, the remainder is 8.

Step 3: Now let us bring down 75, making it the new dividend. Add the previous divisor with itself, 5 + 5, which results in 10, our new divisor.

Step 4: The new divisor will be 10n where n is such that 10n x n is less than or equal to 875. We find n = 8, as 10 x 8 x 8 = 800.

Step 5: Subtract 800 from 875, the remainder is 75. The quotient is now 58.

Step 6: Since the remainder is still present, add a decimal point and bring down two zeros, making the dividend 7500.

Step 7: The new divisor is 116, because 1168 x 8 = 9344, continuing this process until the desired decimal places are reached. So the square root of √3375 is approximately 58.094.

The approximation method is useful for finding square roots easily. Let us learn how to find the square root of 3375 using the approximation method.

Step 1: Find the closest perfect squares to √3375. The largest perfect square smaller than 3375 is 3249, and the smallest perfect square larger than 3375 is 3481. √3375 falls between 57 and 59.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Using the formula: (3375 - 3249) / (3481 - 3249) = 0.126 Add this decimal to the smaller estimate: 57 + 0.126 = 57.126, so the approximate square root of 3375 is 58.094.

• 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, so √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 positive square root is often more relevant in practical applications, which is why it is also known as the principal square root.

• Prime factorization: Breaking down a number into its basic building blocks, specifically prime numbers. For example, the prime factorization of 3375 is 3^4 x 5^3.

• Long division method: A technique used to find the square root of a non-perfect square by dividing the number into groups and using a systematic approach to approximate the square root.