Square Root of 3225

The square root is the inverse of the square of the number. 3225 is not a perfect square. The square root of 3225 is expressed in both radical and exponential form. In the radical form, it is expressed as √3225, whereas in the exponential form it is expressed as (3225)^(1/2). √3225 ≈ 56.775, 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 3225 is broken down into its prime factors.

Step 1: Finding the prime factors of 3225 Breaking it down, we get 3 x 5 x 5 x 43: 3^1 x 5^2 x 43^1

Step 2: Now we found out the prime factors of 3225. The second step is to make pairs of those prime factors. Since 3225 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 3225 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 3225, we need to group it as 32 and 25.

Step 2: Now we need to find n whose square is closest to 32. We can say n as ‘5’ because 5 x 5 = 25 is lesser than 32. Now the quotient is 5 and after subtracting 25 from 32, the remainder is 7.

Step 3: Now let us bring down 25 which is the new dividend. Add the old divisor with the same number 5 + 5 to get 10, which will be our new divisor.

Step 4: The new divisor will be 10n. We need to find the value of n such that 10n x n ≤ 725. Let us consider n as 7, now 107 x 7 = 749, which is more than 725, so n should be less than 7.

Step 5: If n = 6, then 106 x 6 = 636, which fits. Subtracting gives a remainder of 89.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point and add two zeroes to the dividend. Now the new dividend is 8900.

Step 7: Now we need to find the new divisor. Extend the divisor to 1066 and find n such that 1066n x n is close to 8900. Let n = 8, then 10668 x 8 = 85344, which is an overestimate, thus n should be less than 8.

Step 8: Try n = 7, then 10667 x 7 = 74669, which fits. Subtracting gives a remainder of 1431.

Step 9: Continue doing these steps until we get two numbers after the decimal point. For example, the result will approximate to 56.77. So the square root of √3225 is approximately 56.77.

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 3225 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √3225. The closest perfect square less than 3225 is 3136 (56^2) and the closest perfect square greater than 3225 is 3364 (58^2). √3225 falls somewhere between 56 and 58.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (3225 - 3136) ÷ (3364 - 3136) = 89 ÷ 228 ≈ 0.39035. 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 56 + 0.39 ≈ 56.39. Therefore, the square root of 3225 is approximately 56.39.

• 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 a principal square root.

• 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.

• Perfect square: A perfect square is an integer that is the square of another integer. For example, 9 is a perfect square because it is 3^2.