Square Root of 30625

The square root is the inverse of the square of a number. 30625 is a perfect square. The square root of 30625 can be expressed in both radical and exponential form. In the radical form, it is expressed as √30625, whereas (30625)^(1/2) in the exponential form. √30625 = 175, which is a rational number because it can 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. For non-perfect square numbers, the long-division method and approximation method are used. Let's 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 30625 is broken down into its prime factors:

Step 1: Finding the prime factors of 30625 Breaking it down, we get 5 x 5 x 5 x 5 x 7 x 7: (5^4) x (7^2)

Step 2: Now we found the prime factors of 30625. The second step is to make pairs of those prime factors. Since 30625 is a perfect square, each prime factor can be paired.

Therefore, the square root of 30625 is the product of one factor from each pair, which is 5 x 5 x 7 = 175.

The long division method is particularly used for non-perfect square numbers but can also confirm the square root of perfect squares. In this method, we 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 30625, we group it as 30 625.

Step 2: Now, we need to find n whose square is 30 or less. We can say n is ‘5’ because 5 x 5 = 25, which is less than 30. Now the quotient is 5, and after subtracting 30 - 25, the remainder is 5.

Step 3: Bring down 625, making the new dividend 5625. Add the old divisor with the quotient, 5 + 5 = 10, which becomes the new divisor.

Step 4: Find n such that (10n) x n ≤ 5625. Here, n is 5 because (105) x 5 = 525. Step 5: Subtract 5625 from 525, leaving a remainder of 0.

Since the remainder is 0, the quotient 175 is the square root of 30625.

The approximation method is useful for finding the square roots, especially for non-perfect squares. However, for perfect squares like 30625, this method confirms the value.

Step 1: Find the closest perfect squares around 30625. The smallest perfect square is 30276 (174^2) and the largest is 30976 (176^2). √30625 falls between 174 and 176.

Step 2: Since 30625 is closer to the middle of these two squares, the square root is exactly 175.

• Square root: A square root is the inverse of squaring a number. For example: 15^2 = 225, so the square root of 225 is √225 = 15.

• Perfect square: A number that is the square of an integer. For example, 30625 is a perfect square because 175^2 = 30625.

• Rational number: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. For example, 175 is a rational number.

• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 30625 is 5^4 x 7^2.

• Long division method: A method used to find the square root of a number by dividing it into manageable parts.