Square Root of 87

The square root is the inverse of the square of the number. 87 is not a perfect square. The square root of 87 is expressed in both radical and exponential form.

In the radical form, it is expressed as √87, whereas (87)^(1/2) in the exponential form. √87 ≈ 9.327, 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, the prime factorization 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:

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 87 is broken down into its prime factors.

Step 1: Finding the prime factors of 87 Breaking it down, we get 3 x 29.

Step 2: Now we found out the prime factors of 87. The second step is to make pairs of those prime factors. Since 87 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 87 using prime factorization is not possible for finding an exact integer result.

The long division method is particularly used for non-perfect square numbers. 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 87, we do not need to group since it is a two-digit number.

Step 2: Now we need to find n whose square is less than or equal to 87. We can say n as ‘9’ because 9 x 9 = 81, which is less than 87. Now the quotient is 9, and the remainder is 87 - 81 = 6.

Step 3: Since the dividend is less than the divisor, add a decimal point to the quotient and bring down a pair of zeroes to the remainder. Now the new dividend is 600.

Step 4: Double the quotient (9) and write it as 18. Now find a digit x such that 18x x x is less than or equal to 600. Let's try x = 3, giving us 183 x 3 = 549.

Step 5: Subtract 549 from 600, and the remainder is 51. The quotient is now 9.3.

Step 6: Continue the process by bringing down more pairs of zeroes until you achieve the desired decimal precision.

So the square root of √87 is approximately 9.327.

The approximation method is an easy method to find the square root of a given number. Now let us learn how to find the square root of 87 using the approximation method.

Step 1: Identify the closest perfect squares around 87. The nearest perfect squares are 81 (9^2) and 100 (10^2). √87 falls between 9 and 10.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (87 - 81) / (100 - 81) = 6 / 19 ≈ 0.316

Adding this to the smaller perfect square root gives us 9 + 0.316 = 9.316,

so the square root of 87 is approximately 9.316.

• Square root: A square root is the inverse of a square. Example: 4² = 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.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups of two digits from right to left.

• Prime factorization: The process of breaking down a number into its basic prime number factors, which is useful for simplifying radicals.