Square Root of 520

The square root is the inverse of the square of the number. 520 is not a perfect square. The square root of 520 is expressed in both radical and exponential form. In the radical form, it is expressed as √520, whereas in the exponential form it is expressed as (520)^(1/2). √520 ≈ 22.8035, 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 prime factorization method is not used; instead, methods like the long division method and approximation method are employed. Let us now learn these methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 520 is broken down into its prime factors.

Step 1: Finding the prime factors of 520 Breaking it down, we get 2 x 2 x 2 x 5 x 13: 2^3 x 5 x 13

Step 2: Now we found out the prime factors of 520. The second step is to make pairs of those prime factors. Since 520 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 520 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: To begin with, we need to group the numbers from right to left. In the case of 520, we group it as 20 and 5.

Step 2: Now, we need to find n whose square is ≤ 5. We can say n as '2' because 2 x 2 = 4, which is less than or equal to 5. Now the quotient is 2, and subtracting 4 from 5 gives us a remainder of 1.

Step 3: Bring down 20, making the new dividend 120. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find n such that 4n x n ≤ 120. Let us consider n as 2, now 42 x 2 = 84.

Step 5: Subtract 84 from 120, the difference is 36. The quotient is 22.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 3600.

Step 7: We need to find the new divisor. Let it be 446 because 446 x 6 = 2676.

Step 8: Subtracting 2676 from 3600 gives us a remainder of 924.

Step 9: The quotient is now 22.8.

Step 10: Continue with these steps until the desired decimal places are reached.

So the square root of √520 is approximately 22.80.

The approximation method is an easy method for finding the square roots of numbers. Let us learn how to find the square root of 520 using the approximation method.

Step 1: Find the closest perfect square numbers to √520. The smallest perfect square less than 520 is 484 (22²) and the largest perfect square greater than 520 is 529 (23²). √520 falls somewhere between 22 and 23.

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

Using the formula (520 - 484) / (529 - 484) = 36 / 45 = 0.8. Adding this to 22 gives 22 + 0.8 = 22.8,

so the square root of 520 is approximately 22.8.

• Square root: A square root is the inverse of a square. For example, 5² = 25, and the inverse of the square is the square root, which is √25 = 5.

• 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

• .Principal square root: A number has both positive and negative square roots, but the positive square root is often more prominent due to its practical applications, making it the principal square root.

• Prime factorization: Breaking down a number into its prime factors, which are numbers that have only two divisors: 1 and themselves.

• Decimal: A decimal is a number that consists of a whole number and a fraction written together, for example, 7.86, 8.65, and 9.42 are decimals.