Square Root 84: Discover the Calculation and Applications

The square root of 84 is approximately \( \sqrt{84} \approx 9.165 \).

The square root of 84, denoted as \( \sqrt{84} \), is a mathematical operation that determines a number which, when multiplied by itself, equals 84. Here's a step-by-step breakdown:

1. Start with the number 84.
2. Find two perfect square numbers between which 84 lies. For 84, these are \( 9^2 = 81 \) and \( 10^2 = 100 \).
3. Estimate \( \sqrt{84} \) to be between 9 and 10.
4. Refine the estimate using the average: \( \sqrt{84} \approx \frac{9 + 10}{2} = 9.165 \).
5. Verify by squaring 9.165: \( 9.165^2 \approx 84 \).

This process demonstrates how \( \sqrt{84} \) is calculated and its relevance in mathematical contexts.

To calculate \( \sqrt{84} \), follow these steps:

1. Identify the nearest perfect squares surrounding 84: \( 9^2 = 81 \) and \( 10^2 = 100 \).
2. Estimate \( \sqrt{84} \) to be between 9 and 10.
3. Use the average to refine the estimate: \( \sqrt{84} \approx \frac{9 + 10}{2} = 9.5 \).
4. Check by squaring 9.5: \( 9.5^2 = 90.25 \), which is close to 84.
5. Further refine the calculation: \( \sqrt{84} \approx 9.165 \).

This method outlines the precise steps to determine \( \sqrt{84} \) using basic mathematical principles.

The square root of 84, denoted as √84, is calculated as follows:

1. Start with an initial guess. For √84, a reasonable estimate is √81 = 9.
2. Improve the estimate using the formula: \( x_{n+1} = \frac{1}{2} \left( x_n + \frac{84}{x_n} \right) \).
3. Apply the iterative process until convergence:

n	x_n
0	9 (initial guess)
1	\( \frac{1}{2} \left( 9 + \frac{84}{9} \right) = 9.3333 \)
2	\( \frac{1}{2} \left( 9.3333 + \frac{84}{9.3333} \right) = 9.1650 \)
3	\( \frac{1}{2} \left( 9.1650 + \frac{84}{9.1650} \right) = 9.1650 \)

Therefore, the square root of 84, \( \sqrt{84} \approx 9.1650 \).

The square root of 84, denoted as \( \sqrt{84} \), finds applications in various mathematical contexts:

1. Geometry: \( \sqrt{84} \) can be used to calculate the diagonal of a square whose side length is 42 units.
2. Algebra: It appears in algebraic expressions involving simplification and factorization.
3. Number Theory: \( \sqrt{84} \) is involved in discussions about divisibility, factors, and prime factorization of the number 84.
4. Approximation: It serves as a useful approximation in calculations where the exact value is not needed but a close estimate suffices.