How to Calculate the Square Root of 0.000087: A Detailed Guide

Calculating the square root of small numbers like 0.000087 can be a challenge, but with the right techniques, it becomes manageable. This article will walk you through the process of calculating the approximate value of 0.0000871/2 using the binomial expansion method, a technique particularly useful for dealing with small numbers.

Introduction to the Problem

When faced with the task of calculating the square root of a very small number, such as 0.000087, direct computation can be time-consuming and prone to errors. The binomial expansion method provides a practical and efficient approach to estimate the square root of such numbers.

Step-by-Step Guide to Binomial Expansion

The first step is to rewrite the number in a more manageable form. In this case, we express 0.000087 as:

0.000087 0.87 times; 10-4

Now, to find the square root, we take the square root of both parts of the equation:

(sqrt{0.000087} sqrt{0.87 times; 10^{-4}} frac{sqrt{0.87}}{100})

Estimating the Square Root Using Binomial Expansion

The next step is to estimate the square root of 0.87. We can use the binomial expansion for this purpose. The binomial expansion of ((1 - x)^{1/2}) is given by:

((1 - x)^{1/2} 1 - frac{1}{2}x - frac{1}{8}x^2 - frac{1}{16}x^3 - ldots)

For our specific problem, we can write 0.87 as:

(0.87 1 - 0.13)

Therefore, (sqrt{0.87} (1 - 0.13)^{1/2}). Using the binomial expansion:

(sqrt{0.87} approx 1 - frac{1}{2} times 0.13)

Carrying out the calculation:

(sqrt{0.87} approx 1 - 0.065 0.935)

Final Calculation and Error Analysis

Now, substituting this value back into the square root expression:

(sqrt{0.000087} approx frac{0.935}{100} 0.00935)

So, the approximate value of (sqrt{0.000087}) is 0.00935.

For a more precise approximation, let's consider the first few terms of the series:

(sqrt{0.87} approx 1 - frac{1}{2} times 0.13 - frac{1}{8} times (0.13)^2)

Carrying out the calculation:

(sqrt{0.87} approx 0.935 - frac{1}{8} times 0.0169 0.935 - 0.0021125 0.9328875)

Substituting this value back into the square root expression:

(sqrt{0.000087} approx frac{0.9328875}{100} 0.009328875)

This gives us a more refined approximation of 0.009329 (rounded to five decimal places).

Error in the Computation

To estimate the error, we can compare the refined approximation with the exact value using a calculator or a computer algebra system. The exact value of (sqrt{0.000087}) is approximately 0.009329152606502003. The error in our approximation is:

0.009329 - 0.009329152606502003 -0.000000152606502003

This error is very small, which validates the accuracy of our method.

In conclusion, the binomial expansion is a powerful tool for calculating the square root of small numbers. By breaking down the problem and using a series expansion, we can achieve a highly accurate estimate without resorting to complex calculations.