Square Root of 306.25

The square root is the inverse of the square of the number. 306.25 is not a perfect square. The square root of 306.25 is expressed in both radical and exponential form. In the radical form, it is expressed as √306.25, whereas (306.25)^(1/2) in exponential form. √306.25 = 17.5, 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. However, the prime factorization method is not typically used for non-perfect square numbers where long-division method and approximation method are generally applied. 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 306.25 is broken down into its prime factors.

Step 1: Converting 306.25 to a fraction for easier factoring gives us 30625/100.

Step 2: Finding the prime factors of 30625, we have 5^2 × 1225. Further factoring 1225, we get 5^2 × 7^2. So, 30625 = 5^4 × 7^2.

Step 3: The prime factorization of 306.25 in decimal form involves taking the square root of each prime factor pair. √(5^4 × 7^2) = 5^2 × 7 = 25 × 7 = 175. Since we are working with a fraction (30625/100), we need to take the square root of the denominator as well: √100 = 10.

The final result is 175/10 = 17.5.

The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, pair the digits of 306.25 from right to left as 06|25|00.

Step 2: Find the largest number whose square is less than or equal to the first pair (06). That number is 2. Place 2 above the line.

Step 3: Subtract 4 (2^2) from 6, bringing down the next pair to get 225.

Step 4: The divisor is 4 now, and placing 7 next to it gives 47. Find a digit n such that 47n × n ≤ 225.

Step 5: n = 5 as 475 × 5 = 225. Subtract to get 0, and bring down the next pair, 00.

Step 6: The next divisor is 50, placing 0 next to it gives 500.

Since we have no remainder, the square root is 17.5.

The approximation method is another method for finding the square roots; it is an easy way to find the square root of a given number. Now let us learn how to find the square root of 306.25 using the approximation method.

Step 1: Find the closest perfect squares surrounding 306.25. The smallest perfect square is 289 (17^2), and the largest perfect square is 324 (18^2).

Step 2: Since 306.25 is closer to 289, we approximate its square root as between 17 and 18.

Step 3: Calculate the decimal point using interpolation. The formula is: (Given number - smallest perfect square) / (Difference between perfect squares) (306.25 - 289) / (324 - 289) = 17.5 The final approximation is 17.5, so the square root of 306.25 is 17.5.

• 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.

• Rational number: A rational number is a number that can be written in the form of p/q, 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 the principal square root.

• Interpolation: Interpolation is a method of estimating unknown values that fall between known values.

• Factorization: Factorization is the process of breaking down numbers into their constituent prime factors.