Square Root of 725

The square root is the inverse of squaring a number. 725 is not a perfect square. The square root of 725 is expressed in both radical and exponential forms. In the radical form, it is expressed as √725, whereas in exponential form it is (725)^(1/2). √725 = 26.92582, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long division method and approximation method are used. Let us now learn these 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 725 is broken down into its prime factors.

Step 1: Finding the prime factors of 725.

Breaking it down, we get 5 x 5 x 29: 5^2 x 29^1.

Step 2: Now we have found the prime factors of 725. The second step is to make pairs of those prime factors. Since 725 is not a perfect square, the digits of the number can’t be grouped into complete pairs. Therefore, calculating √725 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 numbers for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Begin by grouping the numbers from right to left. In the case of 725, we need to group it as 25 and 7.

Step 2: Now we need to find n whose square is ≤ 7. We can say n as ‘2’ because 2 x 2 = 4 is less than 7. Now the quotient is 2, after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

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

Step 5: The next step is finding 4n × n ≤ 325. Let us consider n as 7, now 47 x 7 = 329.

Step 6: Subtract 325 from 329, but since 329 is too large, try a smaller number. Use 46 x 6 = 276. Subtract 276 from 325 and the difference is 49, and the quotient is 26.

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 4900.

Step 8: Now we need to find the new divisor. Try 539 since 539 x 9 = 4851.

Step 9: Subtracting 4851 from 4900 gives the result 49.

Step 10: Now the quotient is 26.9.

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

So the square root of √725 is approximately 26.93.

Approximation 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 725 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √725. The smallest perfect square less than 725 is 676 (26^2) and the largest perfect square greater than 725 is 729 (27^2). √725 falls between 26 and 27.

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 (725 - 676) ÷ (729 - 676) = 49 ÷ 53 ≈ 0.9245. 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 26 + 0.9245 ≈ 26.925, so the square root of 725 is approximately 26.93.

• Square root: A square root is the inverse operation to squaring a number. Example: 5^2 = 25 and the inverse of the square is the square root, √25 = 5.

• Irrational number: An irrational number is a number that cannot be written in the form of a simple fraction (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, but in most contexts, the positive square root is used, known as the 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.

• Prime factorization: It is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 725 is 5 x 5 x 29.