Square Root of 870

The square root is the inverse of the square of a number. 870 is not a perfect square. The square root of 870 is expressed in both radical and exponential form. In the radical form, it is expressed as √870, whereas (870)^(1/2) in the exponential form. √870 ≈ 29.4958, 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 870 is broken down into its prime factors:

Step 1: Finding the prime factors of 870

Breaking it down, we get 2 x 3 x 5 x 29: 2^1 x 3^1 x 5^1 x 29^1

Step 2: Now we have found the prime factors of 870. The second step is to make pairs of those prime factors. Since 870 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √870 using prime factorization alone 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, group the numbers from right to left. In the case of 870, we need to group it as 70 and 8.

Step 2: Now we need to find a number whose square is less than or equal to 8. We can say this number is 2, as 2 x 2 = 4. The quotient is 2, and the remainder is 8 - 4 = 4.

Step 3: Bring down 70, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: Find a digit n such that 4n x n is less than or equal to 470. Let n be 9, then 49 x 9 = 441.

Step 5: Subtract 441 from 470, the difference is 29, and the quotient becomes 29.

Step 6: Since the dividend is less than the divisor, add a decimal point. Now add two zeroes to the dividend, making it 2900.

Step 7: Find the new divisor, which is 298, as 298 x 9 = 2682.

Step 8: Subtract 2682 from 2900 to get a result of 218.

Step 9: The process continues, and the quotient is approximately 29.49.

So, the square root of √870 is approximately 29.4958.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let's learn how to find the square root of 870 using the approximation method.

Step 1: Find the closest perfect square of √870. The smallest perfect square less than 870 is 841, and the largest perfect square greater than 870 is 900. √870 falls somewhere between 29 and 30.

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

Using the formula: (870 - 841) / (900 - 841) = 29/59 ≈ 0.49. Adding this to 29, the square root of 870 is approximately 29.49.

• 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 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.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it equals 4^2.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 28 is 2 x 2 x 7.