Square Root of 9.5

The square root is the inverse of the square of the number. 9.5 is not a perfect square. The square root of 9.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √9.5, whereas (9.5)^(1/2) in the exponential form. √9.5 ≈ 3.0822, 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 generally used for perfect square numbers. However, for non-perfect square numbers such as 9.5, 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. 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: Group the digits of the number, starting from the decimal point. In the case of 9.5, consider it as 95 with an implied decimal after the 9.

Step 2: Find a number whose square is less than or equal to 9. In this case, 3^2 = 9. The quotient is 3.

Step 3: Subtract 9 from 9 to get 0, and bring down the next pair, which is 50.

Step 4: Double the current quotient (3) to get 6. Find a digit x such that 6x * x is less than or equal to 50. In this case, x=0 works as 60 * 0 = 0.

Step 5: Subtract 0 from 50 to get 50. Since we need more precision, bring down two more zeros to make it 5000.

Step 6: The new divisor becomes 60. Find a digit x such that 60x * x is less than or equal to 5000. Here, x=8 works as 608 * 8 = 4864.

Step 7: Subtract 4864 from 5000 to get 136. The quotient is now approximately 3.08. Continue with these steps until you reach the desired decimal precision.

The square root of 9.5 is approximately 3.0822.

The approximation method is an easy way to find the square root of a given number. Let us learn how to find the square root of 9.5 using the approximation method.

Step 1: Identify the perfect squares between which 9.5 falls. The smallest is 9 (3^2) and the largest is 16 (4^2). Therefore, √9.5 falls between 3 and 4.

Step 2: Use the formula: (Given number - smallest perfect square) / (Next perfect square - smallest perfect square). Applying this, (9.5 - 9) / (16 - 9) = 0.5 / 7 = 0.0714.

Step 3: Add this result to the lower bound of the range, which is 3 + 0.0714 ≈ 3.0714.

Thus, the approximate square root of 9.5 is 3.0714.

• Square root: A square root is the inverse of squaring a number. For example, 3^2 = 9, and the square root of 9 is √9 = 3.
   
• Irrational number: An irrational number is a number that cannot be written as a simple fraction, such as the square root of a non-perfect square.
   
• Decimal: A decimal is a number that has a whole number part and a fractional part separated by a decimal point, such as 9.5.
   
• Long division method: A method used to find the square roots of non-perfect squares by performing step-by-step division.
   
• Approximation: A method of finding a value that is close to, but not exactly equal to, the square root of a number, typically by using nearby perfect squares.