Square Root of 10.4

The square root is the inverse of the square of the number. 10.4 is not a perfect square. The square root of 10.4 is expressed in both radical and exponential form. In the radical form, it is expressed as √10.4, whereas (10.4)^(1/2) in the exponential form. √10.4 ≈ 3.2249, 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 long-division and approximation methods 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. 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 10.4, we need to consider it as 10.40.

Step 2: Now, find n whose square is less than or equal to 10. The closest perfect square is 9, which gives n as 3. Now, the quotient is 3, and the remainder is 1.

Step 3: Bring down the next pair of digits (40) to make the new dividend 140. Add the old divisor with the same number, 3 + 3, to get 6, which will be our new divisor.

Step 4: The new divisor is 6n. Find n such that 6n × n ≤ 140. Let n be 2, then 62 × 2 = 124.

Step 5: Subtract 124 from 140, the difference is 16, and the quotient is 3.2.

Step 6: Since we need more precision, add another pair of zeros to the remainder to make it 1600.

Step 7: The previous divisor was 62, so now it becomes 64n. Find n such that 64n × n ≤ 1600. Let n be 2, then 642 × 2 = 1284.

Step 8: Subtract 1284 from 1600, and the difference is 316. The quotient is 3.22.

Step 9: Repeat these steps until the desired precision is achieved.

The square root of 10.4 is approximately 3.2249.

The approximation method is an easy method to find the square root of a given number. Now let us learn how to find the square root of 10.4 using the approximation method.

Step 1: We need to find the closest perfect squares to √10.4. The smallest perfect square less than 10.4 is 9, and the closest perfect square greater than 10.4 is 16. √10.4 falls somewhere between 3 and 4.

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

Using the formula, (10.4 - 9) / (16 - 9) = 1.4 / 7 = 0.2.

Adding this to the smaller integer value of the square root, 3 + 0.2 = 3.2.

Therefore, the square root of 10.4 is approximately 3.2.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, since 4 × 4 = 16.
   
• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction. Its decimal goes on forever without repeating. For example, √10.4 is an irrational number.
   
• Principal square root: A number has both positive and negative square roots, but the principal square root is the non-negative root, typically used in calculations.
   
• Long division method: A technique used to find the square roots of imperfect squares, involving repeated division and averaging.
   
• Decimal: A number that contains a whole number and a fractional part separated by a decimal point, such as 3.2249.