Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Roots Evaluate each radical by calculator, retaining the proper number of digits in your answer: $$\sqrt{1.863}$$

Short answer

The square root of 1.863 rounded to four significant digits is approximately 1.364.

Step-by-step solution

Understanding the Problem

We are asked to find the square root of the number 1.863. It's important to note that we must retain the proper number of significant digits in our answer. The number 1.863 has four significant digits.

Calculate the Square Root

Use a calculator to find the square root of 1.863. This should be done to at least four significant digits to match the original number.

Retain Significant Digits

After calculating, round your answer to four significant digits. This is done because the square root operation does not affect the number of significant digits.

Key concepts

Significant Digits

Understanding and correctly applying the concept of significant digits, or significant figures, is crucial when working with measurements and calculations in science and mathematics. These are the digits in a number that carry meaning contributing to its accuracy. They include all non-zero numbers, any zeros between significant digits, and any trailing zeros in a decimal quantity.

For example, in the number 1.863, there are four significant digits: 1, 8, 6, and 3. The zeros that might follow the last three as in 1.8630 are also significant as they are in a decimal and indicate precision. When performing calculations, it's essential to carry through an appropriate number of significant digits to ensure that the final answer reflects the precision of the most accurately known number. This becomes particularly important when working with square root calculations, where the resulting value should retain the same level of precision as the initial value.

Square Root Calculation

Calculating the square root of a number involves finding a value that, when multiplied by itself, yields the original number. This can be a straightforward process with the use of calculators or software designed to compute square roots. However, the challenge often lies in retaining the accuracy and precision of the original number, which is where significant digits come into play.

In the provided exercise, finding the square root of 1.863 should be approached with the understanding that the precision of the outcome should not exceed that of our initial number. Since 1.863 has four significant figures, the square root must be approximated to four significant figures as well. Remember to check your calculator settings to ensure it displays an adequate number of digits for complete accuracy.

Radical Expression Evaluation

The evaluation of radical expressions, including square roots, involves not just finding the numerical value but also appreciating the precision of the result. When the calculator provides a result for the square root of 1.863, it might show a long string of digits. However, retaining the proper number of significant figures from the original number is key.

After getting the initial result, the next step is to round it to four significant digits. This may involve rounding up or down the last digit considered significant, according to standard rounding rules. For example, if our calculator shows a result of 1.364355..., we would round this to 1.364 as the fourth digit after the decimal is less than 5. Thus, 1.364 would be our final answer when evaluating the square root of 1.863 with the correct number of significant digits.