Square Root of 1828

The square root is the inverse of squaring a number. 1828 is not a perfect square. The square root of 1828 can be expressed in both radical and exponential forms. In radical form, it is expressed as √1828, whereas in exponential form as (1828)^(1/2). √1828 ≈ 42.7465, which is an irrational number because it cannot be expressed as a fraction of two integers.

The prime factorization method is used for perfect square numbers. For non-perfect square numbers, methods like long division and approximation are used. Let us now learn these 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 1828 is broken down into its prime factors:

Step 1: Finding the prime factors of 1828 Breaking it down, we get 2 x 2 x 457: 2^2 x 457

Step 2: Now we found out the prime factors of 1828. Since 1828 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating the square root of 1828 using prime factorization directly is not possible.

The long division method is particularly useful 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, group the numbers from right to left. In the case of 1828, group it as 18 and 28.

Step 2: Find n whose square is less than or equal to 18. We can say n is ‘4’ because 4 x 4 = 16, which is less than 18. The quotient is 4, and after subtracting 16 from 18, the remainder is 2.

Step 3: Bring down 28, creating a new dividend of 228. Add the old divisor with itself: 4 + 4 = 8, which will be our new divisor.

Step 4: Find n such that 8n x n is less than or equal to 228. Let n be 2, then 82 x 2 = 164.

Step 5: Subtract 164 from 228, resulting in 64. The quotient is 42.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros. The new dividend is 6400.

Step 7: Find the new divisor. Adding another 2 to 84 gives 842. Now find n such that 842n x n is less than or equal to 6400. Let n be 7, then 842 x 7 = 5894.

Step 8: Subtract 5894 from 6400, resulting in 506.

Step 9: The quotient is approximately 42.7.

Step 10: Continue these steps until you achieve the desired decimal precision.

Thus, the square root of √1828 is approximately 42.7465.

The approximation method is another approach for finding square roots, providing an easy way to estimate the square root of a given number. Let us learn how to find the square root of 1828 using the approximation method.

Step 1: Identify the closest perfect squares around 1828. The smallest perfect square is 1764 (42^2) and the largest is 1849 (43^2). √1828 falls between 42 and 43.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula: (1828 - 1764) / (1849 - 1764) = 64 / 85 ≈ 0.7529

Adding this decimal to the smaller square root: 42 + 0.7529 ≈ 42.7529

Therefore, the square root of 1828 is approximately 42.7529.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse is √16 = 4.

• Irrational number: An irrational number cannot be expressed as a fraction of two integers, where the denominator is not zero.

• Approximation method: A technique used to estimate the value of a square root by identifying nearby perfect squares and using interpolation.

• Long division method: A step-by-step division process used to find the square root of non-perfect squares.

• Prime factorization: Breaking down a number into its prime factors, which are used to determine properties like square roots.