Square Root of 86

The square root is the inverse of the square of the number. 86 is not a perfect square. The square root of 86 is expressed in both radical and exponential form.

In radical form, it is expressed as √86, whereas (86)^(1/2) in the exponential form. √86 ≈ 9.2736, 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 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 86 is broken down into its prime factors.

Step 1: Finding the prime factors of 86 Breaking it down, we get 2 x 43: 2¹ x 43¹

Step 2: Now we have found the prime factors of 86. The second step is to make pairs of those prime factors. Since 86 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 86 using prime factorization is impossible.

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 86, we need to group it as 86.

Step 2: Now we need to find n whose square is ≤ 86. We can say n as ‘9’ because 9 x 9 = 81, which is less than 86. The quotient is 9, and after subtracting 81 from 86, the remainder is 5.

Step 3: Since there are no more digits to bring down, we need to add a decimal point and bring down two zeros to make the dividend 500.

Step 4: Double the previous quotient (9) to get 18, which will be our new divisor prefix. We need to find a digit x such that 18x x x ≤ 500.

Step 5: Let x be 2, then 182 x 2 = 364.

Step 6: Subtract 364 from 500 to get 136, and the quotient is now 9.2.

Step 7: Since the remainder is not zero, we repeat the process by bringing down more pairs of zeros and finding the next digit. Continue until you reach the desired precision.

So, the square root of √86 is approximately 9.2736.

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 us learn how to find the square root of 86 using the approximation method.

Step 1: Find the closest perfect squares to 86. The smallest perfect square less than 86 is 81, and the largest perfect square greater than 86 is 100. √86 falls somewhere between 9 and 10.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (86 - 81) / (100 - 81) = 5 / 19 ≈ 0.263.

Step 3: Add the decimal to the smaller root: 9 + 0.263 = 9.263.

So, the square root of 86 is approximately 9.2736.

• Square root: A square root is the inverse operation of squaring a number. For example, 3² = 9, so the square root of 9 is 3: √9 = 3.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction, meaning it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.

• Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 25 is 5.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 86 is 2 x 43.

• Decimal: A decimal number is a number that consists of a whole number and a fractional part separated by a decimal point. For example, 9.2736 is a decimal number. ```