Square Root of 3.5

The square root is the inverse of the square of the number. 3.5 is not a perfect square. The square root of 3.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √3.5, whereas (3.5)^(1/2) is in the exponential form. √3.5 ≈ 1.8708, 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. Let us now learn how to find the square root using the long division method, step by step:

Step 1: Start by placing 3.5 under the long division symbol, treating it as 35 to avoid dealing with a decimal initially, then consider it as 3500.

Step 2: Find the largest number whose square is less than or equal to 35. Here, it is 5 because 5 × 5 = 25, which is less than 35.

Step 3: Subtract 25 from 35 to get 10 and bring down the next two zeros, making it 1000.

Step 4: Double the quotient (5) to get 10 as the new divisor.

Step 5: Find a number (n) such that 10n × n ≤ 1000. Here, n = 9 works because 109 × 9 = 981.

Step 6: Subtract 981 from 1000 to get 19 and bring down the next two zeros, making it 1900.

Step 7: Continue this process to get decimal places as needed.

The next steps will eventually give a quotient of about 1.8708.

Approximation is another method for finding the 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 3.5 using the approximation method.

Step 1: Find the closest perfect squares around 3.5. The closest perfect squares are 1 (1^2) and 4 (2^2). Thus, √3.5 falls between 1 and 2.

Step 2: Use linear interpolation to approximate the decimal. The formula is: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). For √3.5: (3.5 - 1) / (4 - 1) = 2.5/3 ≈ 0.8333.

Step 3: Add this decimal to the smaller square root: 1 + 0.8333 = 1.8333, but refine it further through more precise interpolation to get approximately 1.8708.

• Square root: A square root is the inverse of a square. Example: 2^2 = 4, and the inverse of the square is the square root, √4 = 2.

• 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.

• Linear interpolation: A method of estimating unknown values that lie between known values. It's often used for approximating irrational square roots.

• Decimal: If a number has a whole number and a fractional part, it is called a decimal. For example: 3.5, 7.86, and 9.42 are decimals.

• Rational number: A rational number is a number that can be expressed as a fraction, where both the numerator and the denominator are integers and the denominator is not zero. For example: 3.5 can be written as 7/2.