Square Root of 10.5

The square root is the inverse of the square of the number. 10.5 is not a perfect square. The square root of 10.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √10.5, whereas (10.5)^(1/2) in the exponential form. √10.5 ≈ 3.24037, 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:

• Long division method
• Approximation method

Since 10.5 is not a perfect square, the prime factorization method is not applicable. Therefore, we use other methods such as the long division method or approximation to find the square root.

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 consider 10.5 as 10.50 to make grouping easier. Group the numbers from right to left: 10 and 50

Step 2: Now we need to find n whose square is less than or equal to 10. We can say n is '3' because 3 × 3 = 9 is less than 10. Now the quotient is 3, after subtracting 10 - 9, the remainder is 1.

Step 3: Now let us bring down 50, making the new dividend 150. Add the old divisor with the same number 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be 6n. We need to find n such that 6n × n ≤ 150. Let us consider n as 2; now 62 × 2 = 124.

Step 5: Subtract 150 from 124; the difference is 26, and the quotient is 32.

Step 6: Since the dividend is less than the new divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2600.

Step 7: Now we need to find the new divisor, which is 644, because 644 × 4 = 2576.

Step 8: Subtracting 2576 from 2600 gives us 24.

Step 9: Now the quotient is 3.24.

Step 10: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √10.5 is approximately 3.24.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 10.5 using the approximation method.

Step 1: Find the closest perfect squares around 10.5. The smallest perfect square is 9, and the largest perfect square is 16. √10.5 falls somewhere between 3 and 4.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Applying the formula: (10.5 - 9) / (16 - 9) = 1.5 / 7 ≈ 0.2143.

Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number, which is 3 + 0.2143 ≈ 3.2143.

Therefore, the square root of 10.5 is approximately 3.24.

• 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.
   
• 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: A decimal is a number that has a whole number and a fraction in a single number, for example, 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: This is a method used to find the square root of non-perfect square numbers by dividing and averaging over several steps.
   
• Approximation method: A method used to estimate the square root of a number by finding its position between two perfect squares and using proportional calculations.