Square Root of 10.29

The square root is the inverse of the square of the number. 10.29 is not a perfect square. The square root of 10.29 is expressed in both radical and exponential form. In the radical form, it is expressed as √10.29, whereas (10.29)^(1/2) in the exponential form. √10.29 ≈ 3.207, which is an irrational number because it cannot 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 used for non-perfect square numbers where the long-division method and approximation method are used. 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. However, since 10.29 is not a perfect square, calculating its square root using prime factorization is not feasible.

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 10.29, we consider it as 10 and 29.

Step 2: Now we need to find a number n whose square is less than or equal to 10. We can say n is 3 because 3 × 3 = 9, which is less than 10. Subtracting gives a remainder of 1.

Step 3: Bring down 29, making the new dividend 129. Add the old divisor (3) with itself, resulting in 6 as the new divisor.

Step 4: Find the largest digit x such that 6x × x ≤ 129. We find that x is 2 because 62 × 2 = 124, which is less than 129. Subtract to get a remainder of 5.

Step 5: 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, making it 500.

Step 6: The new divisor becomes 64, and we determine the next digit of the quotient.

Repeating similar steps, we get an approximate value for the square root of 10.29 as 3.207.

The approximation method is another method for finding the square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 10.29 using the approximation method.

Step 1: Now we have to find the closest perfect squares surrounding 10.29. The smallest perfect square less than 10.29 is 9, and the largest perfect square greater than 10.29 is 16. So, √10.29 falls somewhere between 3 and 4.

Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (10.29 - 9) / (16 - 9) = 1.29 / 7 ≈ 0.1843 Using the formula, we identified the decimal point of our square root.

Adding the integer part, we get 3 + 0.1843 = 3.1843, so the approximate square root of 10.29 is about 3.207.

• Square root: A square root is the inverse of a square. For example, 4^2 = 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.
   
• 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: An approximation is a value or number that is close to a desired result but not exact, often used when dealing with irrational numbers.
   
• Long Division Method: A mathematical procedure used to find the square root of a number through a series of division steps, particularly useful for non-perfect squares.