Square Root of 824

The square root is the inverse of the square of the number. 824 is not a perfect square. The square root of 824 is expressed in both radical and exponential form.

In radical form, it is expressed as √824, whereas in exponential form, it is expressed as (824)^(1/2). √824 ≈ 28.7228, 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 824 is broken down into its prime factors:

Step 1: Finding the prime factors of 824 Breaking it down, we get 2 x 2 x 2 x 103: 2³ x 103¹

Step 2: Now we have found the prime factors of 824. The next step is to make pairs of those prime factors. Since 824 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √824 using prime factorization is impractical for exact results.

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: To begin with, we need to group the numbers from right to left. In the case of 824, we need to group it as 24 and 8.

Step 2: Now, we need to find a number whose square is less than or equal to 8. We can say n as '2' because 2 x 2 = 4, which is less than 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.

Step 3: Bring down 24, making the new dividend 424. Add the old divisor with the quotient, 2 + 2, to get 4, which will be part of our new divisor.

Step 4: The new divisor is 42n. We need to find the value of n such that 42n x n ≤ 424. Let's consider n as 9, then 42 x 9 = 378.

Step 5: Subtract 378 from 424, the difference is 46, and the quotient is 29.

Step 6: Since there is a remainder, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 4600.

Step 7: Find a new divisor that is 586 because 586 x 8 = 4688.

Step 8: Subtracting 4688 from 4600 gives the result -88.

Step 9: Continue these steps until we get two numbers after the decimal point or until the remainder is zero. So the square root of √824 is approximately 28.72.

The approximation method is another method for finding square roots, and it is an easy method for estimating the square root of a given number. Now let us learn how to find the square root of 824 using the approximation method.

Step 1: Find the closest perfect squares of √824. The smallest perfect square less than 824 is 784 (28²), and the largest perfect square greater than 824 is 841 (29²). √824 falls between 28 and 29.

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

Using the formula, (824 - 784) / (841 - 784) = 40 / 57 ≈ 0.70 Adding this to the smaller perfect square's root gives us 28 + 0.70 = 28.70, so the square root of 824 is approximately 28.72.

• Square root: A square root is the inverse of squaring a number. Example: 4² = 16 and the inverse is the square root, √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. Its decimal goes on forever without repeating.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 1, 4, 9, 16, etc.
   
• Prime factorization: Breaking down a number into its basic building blocks, which are prime numbers.
   
• Approximation: Estimating a number's value close to its true value, often used when precise values are hard to find.