Square Root of 4250

The square root is the inverse of the square of the number. 4250 is not a perfect square. The square root of 4250 is expressed in both radical and exponential form. In the radical form, it is expressed as √4250, whereas (4250)^(1/2) in the exponential form. √4250 ≈ 65.192, 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. Now let us look at how 4250 is broken down into its prime factors.

Step 1: Finding the prime factors of 4250 Breaking it down, we get 2 x 5 x 5 x 17 x 5: 2^1 x 5^3 x 17^1

Step 2: Now we found out the prime factors of 4250. The next step is to make pairs of those prime factors. Since 4250 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 4250 using prime factorization is not straightforward.

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: Begin by grouping the digits from right to left. For 4250, we group it as 50 and 42.

Step 2: Find n whose square is ≤ 42. We can say n is 6 because 6 x 6 = 36 which is less than or equal to 42. Now, the quotient is 6. After subtracting 36 from 42, the remainder is 6.

Step 3: Bring down the next pair of digits, 50, to make the new dividend 650. Add the old divisor with the same number, 6 + 6 = 12, which will be our new divisor.

Step 4: Determine the next digit for the quotient. Consider 12n as the new divisor. We need to find n such that 12n x n ≤ 650. Let n be 5, now 125 x 5 = 625.

Step 5: Subtract 625 from 650; the difference is 25. Since the remainder is less than the divisor, we add a decimal point to the quotient and bring down two zeros, making the new dividend 2500.

Step 6: Consider the new divisor as 130 (after adding the previous quotient digit 5 to 125) and find n such that 130n x n ≤ 2500. Let n be 1, now 130 x 1 = 130.

Step 7: Subtract 130 from 2500 to get 2370. Continue this process to determine more decimal places.

So, the square root of √4250 is approximately 65.192.

The approximation method is another way to find square roots; it is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 4250 using the approximation method.

Step 1: Find the closest perfect squares around 4250. The closest perfect square below 4250 is 4225 (65^2), and the closest perfect square above 4250 is 4356 (66^2). Therefore, √4250 falls between 65 and 66.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (4250 - 4225) / (4356 - 4225) ≈ 0.192 Using the formula, we identified the decimal value of our square root approximation. Add this value to the integer part: 65 + 0.192 = 65.192, so the square root of 4250 is approximately 65.192.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be expressed as a fraction p/q, where q is not zero and p and q are integers. Example: √2.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is typically used in real-world applications, known as the principal square root.
   
• Prime factorization: The expression of a number as a product of prime numbers. Example: The prime factorization of 18 is 2 x 3 x 3.
   
• Decimal approximation: A method to estimate an irrational square root to a certain number of decimal places. Example: √2 ≈ 1.414.