Square Root of 425

The square root is the inverse of the square of a number. 425 is not a perfect square. The square root of 425 is expressed in both radical and exponential form. In radical form, it is expressed as √425, whereas in exponential form, it is expressed as (425)^(1/2). √425 ≈ 20.6155, 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 prime factorization method is not used; instead, the long-division method and approximation method are used. Let us 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 425 is broken down into its prime factors:

Step 1: Finding the prime factors of 425 Breaking it down, we get 5 x 5 x 17: 5² x 17

Step 2: We have found the prime factors of 425. The next step is to make pairs of those prime factors. Since 425 is not a perfect square, the digits of the number can’t be grouped into perfect pairs. Therefore, calculating √425 using prime factorization doesn't yield an exact answer due to the unpaired factor.

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 425, we need to group it as 25 and 4.

Step 2: Now we need to find n whose square is ≤ 4. We can say n is '2' because 2 x 2 = 4. Now the quotient is 2, and after subtracting, the remainder is 0.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number 2 + 2, we get 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 x n ≤ 25. Let us consider n as 5, now 4 x 5 x 5 = 100.

Step 6: Subtract 25 from 100, and the difference is -75, but since 100 exceeds, n needs to be less, so calculate again.

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 2500.

Step 8: Now we need to find the new divisor that is 41 because 415 x 5 = 2075.

Step 9: Subtracting 2075 from 2500, we get the result 425.

Step 10: Now the quotient is 20.5.

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

So the square root of √425 is approximately 20.6155.

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 425 using the approximation method.

Step 1: Now we have to find the closest perfect square of √425. The smallest perfect square less than 425 is 400, and the largest perfect square greater than 425 is 441. √425 falls between 20 and 21.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Using the formula (425 - 400) ÷ (441 - 400) = 25 ÷ 41 ≈ 0.6098.

Using the formula, we identified the decimal part of our square root.

The next step is adding the value we got initially to the decimal number, which is 20 + 0.6098 ≈ 20.61, so the square root of 425 is approximately 20.61.

• Square root: A square root is the inverse of a square. Example: 4² = 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. This is why it is also known as the principal square root.

• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 425 is 5² x 17.

• Approximation: Approximation is the process of finding a value that is close enough to the right answer, usually within a small range of error.