Square Root of 887

The square root is the inverse of the square of a number. 887 is not a perfect square. The square root of 887 is expressed in both radical and exponential form. In radical form, it is expressed as √887, whereas \(887^{1/2}\) in exponential form. √887 ≈ 29.775, 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, this method is not used for non-perfect square numbers, where 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 887 is broken down into its prime factors:

Step 1: Finding the prime factors of 887. Since 887 is not a perfect square and is itself a prime number, it cannot be broken down into simpler prime factors.

Therefore, calculating √887 using prime factorization is not feasible.

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 887, we group it as 87 and 8.

Step 2: Now we need to find n whose square is less than or equal to 8. We can say n as '2' because \(2 \times 2 = 4\), which is less than 8. Now the quotient is 2, after subtracting \(8 - 4\), the remainder is 4.

Step 3: Bring down 87, making the new dividend 487. Add the old divisor (2) with the same number to get 4, which will be part of our new divisor.

Step 4: The new divisor is now 4n. We need to find n such that \(4n \times n \leq 487\). Let \(n = 9\), then \(49 \times 9 = 441\).

Step 5: Subtract 441 from 487, the difference is 46, and the quotient is 29.

Step 6: Since the dividend is less than the divisor, 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 the new divisor, 59, because \(599 \times 9 = 5391\).

Step 8: Subtract 5391 from 4600, resulting in a remainder of 209.

Step 9: Continue doing these steps until we achieve the desired precision.

The square root of √887 is approximately 29.775.

The approximation method is another method for finding square roots and is an easy method to find the square root of a given number. Now let us learn how to find the square root of 887 using the approximation method.

Step 1: Find the closest perfect squares to √887. The smallest perfect square less than 887 is 841, and the largest perfect square greater than 887 is 900. √887 falls somewhere between 29 and 30.

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

Using the formula: \((887 - 841)/(900 - 841) = 46/59 \approx 0.780\).

Thus, the approximate square root of 887 is \(29 + 0.780 = 29.780\).

• Square root: A square root is the inverse of a square. For example, \(4^2 = 16\) and the inverse of the square is the square root, that is, \(\sqrt{16} = 4\).
   
• 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.
   
• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
   
• Decimal: A decimal is a number that has a whole number and a fractional part. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is \(4^2\).