Square root of 84.

The square root of 84 can be easily found out by using long division method. In which it is discovered that the cumulative approximation of √84 is 9.165.
 

There are many ways through which students can find square roots, and some of these methods are very popular. Some of the methods have been explained in detail below.
 

In this method, we decompose the number into its prime factors.

Prime factorization of 84: 84=2×2×3×7

Since not all prime factors can be paired, 84 cannot be simplified into a perfect square. Therefore, the square root of 84 cannot be expressed in a simple radical form.
 

For non-perfect squares, we often use the nearest perfect square to estimate the square root. Follow these steps:

Step 1: Write the number 84 to perform long division.

Step 2: Identify a perfect square number that is less than or equal to 84. For 84, that number is 64 (8²).

Step 3: Divide 84 by 8. The remainder will be 20, and the quotient will be 10.

Step 4: Bring down the remainder (20) and append two zeros. Add a decimal point to the quotient, making it 10.0.

Step 5: Double the quotient to use as the new divisor, which gives 20.

Step 6: Select a number that, when multiplied by the new divisor, results in a product less than or equal to 2000.

Step 7: Continue the division process to find √84 to the desired decimal places. → √84 ≈ 9.165.

In the approximation method, we estimate the square root by identifying the closest perfect squares surrounding the number.

Step 1: The nearest perfect squares to 84 are √100 = 10 and √81 = 9.

Step 2: Since 84 is between 100 and 81, we know the square root will be between 10 and 9.

Step 3: By testing numbers like 9.1, 9.2 and further, we find that √84 ≈ 9.165.
 

• Square Root: A number which when is multiplied by itself gives the original number is called a square root.

• Perfect Square: A number that is the integral square of an integer I such that n = I², example I = 1, 2, 3, n = 1, 4, 9, 16, etc.

• Prime Factorization: The ability to factorize a number in to the product of the basic arithmetic numbers, also known as primary numbers.

• Non-Perfect Square: A figure that cannot be converted into an integer figure once divided by itself (e.g., 76).

• Approximation Method: Approximating square root, that is, finding the closest integer which, when squared, yields the number being approximated.