Square Root of 20.25

The square root is the inverse of the square of the number. 20.25 is a perfect square. The square root of 20.25 is expressed in both radical and exponential form. In the radical form, it is expressed as √20.25, whereas in exponential form it is (20.25)^(1/2). √20.25 = 4.5, 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 can be used for finding the square root of perfect square numbers. Other methods include long division and approximation methods. However, for 20.25, the square root can be directly calculated since it is a perfect square. 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. Below is how 20.25 is broken down into its prime factors:

Step 1: Express 20.25 as a fraction: 20.25 = 2025/100.

Step 2: Find the prime factors of 2025. Breaking it down, we get 3 x 3 x 3 x 3 x 5 x 5: 3^4 x 5^2.

Step 3: Since 2025 is a perfect square, its square root is the product of the square roots of its prime factors: (3^2 x 5)^2 = (3 x 5)^2 = 15^2.

Step 4: Since the square of 15 is 225, we know that the square root of 2025 is 45.

Therefore, the square root of 20.25 is 4.5.

The long division method is particularly used for non-perfect square numbers. However, for a number like 20.25 which is a perfect square, we can find the square root directly or verify using the long division method. Below is a simplified explanation:

Step 1: Start by expressing 20.25 as a decimal number.

Step 2: Group the numbers in pairs from the decimal point: 20, 25.

Step 3: Find a number whose square is less than or equal to the first group. For 20, the closest square is 16 (4^2). Place 4 as the first digit of the square root.

Step 4: Bring down the next pair (25), making it 425.

Step 5: The new divisor is twice the current quotient (4), so it is now 8. Find a digit x such that 8x x is less than or equal to 425. The digit is 5.

Step 6: Multiply 85 by 5 to get 425. Subtract 425 from 425 to get 0. The quotient is 4.5.

Thus, the square root of 20.25 is 4.5.

The approximation method is another method for finding square roots but is not necessary for perfect squares. However, it can be used for understanding:

Step 1: Find the closest perfect squares around 20.25. The closest are 16 (4^2) and 25 (5^2).

Step 2: Since 20.25 is directly between these values, the square root is between 4 and 5. Given its exact position, we can calculate it precisely as 4.5.

Therefore, the square root of 20.25 is approximately 4.5, which is exact due to it being a perfect square.

• 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, where q is not zero and p and q are integers.
   
• Perfect square: A perfect square is a number that has an integer as its square root. For example, 16 is a perfect square because its square root is 4.
   
• 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.
   
• Approximation: Approximation involves estimating the value of a number when precise calculation is not necessary.