Square Root of 34.5

The square root is the inverse of the square of a number. 34.5 is not a perfect square. The square root of 34.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √34.5, whereas (34.5)^(1/2) in the exponential form. √34.5 ≈ 5.87367, 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, for non-perfect square numbers, 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. Since 34.5 is not an integer, direct prime factorization cannot be used. However, if we consider its approximate value, 34.5 can be broken down for estimation purposes:

Step 1: Approximate to the nearest whole number, 35, and find the prime factors. Breaking it down, we get 5 x 7 or 5 x 7.

Step 2: Since 34.5 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating 34.5 using prime factorization for exact value is not possible.

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: To begin with, group the numbers from right to left. For 34.5, consider it as 34.50 for convenience.

Step 2: Find n whose square is closest to 3. The number 1 works because 1 x 1 is less than or equal to 3. The quotient is 1, and the remainder is 2 after subtracting 1 from 3.

Step 3: Bring down 45, making the new dividend 245.

Step 4: Double the previous divisor and place it next to itself (2), making it 2n. Find n such that 2n x n is less than or equal to 245.

Step 5: The value of n is 8 because 28 x 8 = 224.

Step 6: Subtract 224 from 245, leaving a remainder of 21.

Step 7: Add a decimal point and zeros to the dividend. The new dividend is 2100.

Step 8: Continue the process to reach a more accurate value.

The approximate square root of 34.5 is 5.87.

The approximation method is another way to find the square roots, and it is an easy method to find the square root of a given number. Let us learn how to find the square root of 34.5 using the approximation method:

Step 1: Find the closest perfect squares around 34.5. The smallest perfect square less than 34.5 is 25, and the largest perfect square greater than 34.5 is 36. √34.5 falls between √25 = 5 and √36 = 6.

Step 2: Use the interpolation formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (34.5 - 25) / (36 - 25) ≈ 0.8636.

Add this decimal to the lower bound: 5 + 0.8636 ≈ 5.87.

So, the approximate square root of 34.5 is 5.87.

• 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 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.
   
• Approximation: Finding a value that is close to the actual value, often used when exact values are difficult to calculate.
   
• Long division method: A step-by-step approach to find the square root of a non-perfect square, using division and subtraction iteratively.