Square Root of 10.25

The square root is the inverse of the square of the number. 10.25 is a perfect square. The square root of 10.25 is expressed in both radical and exponential form. In the radical form, it is expressed as √10.25, whereas (10.25)^(1/2) in the exponential form. √10.25 = 3.2, 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, for decimal perfect squares, the long-division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation method

The long division method is particularly used for non-perfect square numbers and decimal numbers. In this method, we should check the grouping of digits 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, consider the number 10.25. Place a decimal point and pair the digits before and after the decimal point.

Step 2: Find the largest number whose square is less than or equal to 10. We can choose 3 because 3 × 3 = 9. Subtract 9 from 10, and bring down the next pair of digits, 25, making the new dividend 125.

Step 3: Double the current quotient (3), giving us 6, which will be our new divisor base.

Step 4: Find a digit x such that 6x × x is less than or equal to 125. The suitable digit is 2, making the divisor 62 and the product 62 × 2 = 124.

Step 5: Subtract 124 from 125, leaving a remainder of 1.

Since there are no more pairs of digits, the quotient, 3.2, is the square root of 10.25.

Approximation method is another method for finding square roots; it is an easy method to approximate the square root of a given number. Now let us learn how to find the square root of 10.25 using the approximation method:

Step 1: Identify the perfect squares around 10.25. The closest perfect square less than 10.25 is 9, and the closest perfect square greater than 10.25 is 16.

Step 2: Knowing that √9 = 3 and √16 = 4, we can see that √10.25 is between 3 and 4.

Step 3: Narrow down by approximation. Since 10.25 is closer to 9, we try 3.2. Calculate 3.2 × 3.2 = 10.24, which is very close.

Thus, the square root of 10.25 is approximately 3.2.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: 3.2² = 10.24, and the inverse of the square is the square root, so √10.25 = 3.2.
   
• Rational number: A rational number is a number that can be expressed in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A perfect square is a number that is the square of an integer or a rational number. Example: 10.25 is a perfect square because √10.25 = 3.2.
   
• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A step-by-step process used to find the square root of a number, especially useful for numbers that are not perfect squares or for decimal numbers.