Problem 11 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Powers by Calculator. $$(9.55)^{3}$$

Step-by-Step Solution

Answer

The value of \((9.55)^{3}\) rounded to three significant digits is 871.

Step 1: Identify the Number of Significant Digits

First, determine the number of significant digits in the given number 9.55. Since there are two digits after the decimal point, the number 9.55 has three significant digits.

Step 2: Calculate the Power

Use a calculator to raise 9.55 to the power of 3. When entering the calculation into the calculator, simply type 9.55 followed by the caret symbol for power (^), and then 3.

Step 3: Round to the Proper Number of Significant Digits

After obtaining the result from the calculator, round off your answer to three significant digits. If the calculator gives you a result such as 871.287, you would round it to 871 to retain three significant digits.

Key Concepts

Powers by Calculator

When working with powers and exponents, a calculator is an indispensable tool that simplifies the process significantly. To accurately calculate raised powers like (9.55)^3, you should first ensure your calculator is in the correct mode for handling decimal numbers.
To input the power, type the base number (9.55 in this case), then use the caret symbol (^) or sometimes a dedicated button (like xy), followed by the exponent (3). Press enter, and voilà, you'll see the result on your display. It's important to note, depending on your calculator's make and model, the method for entering exponents may vary slightly.
Before using the result, always remember to check for the number of significant digits to ensure your answer is as precise as required. This neat intersection of technology and mathematical rules allows for rapid and accurate computations.

Rounding Numbers

Rounding is a fundamental technique for managing significant digits in calculations. It helps in simplifying numbers while maintaining the overall accuracy of the result. The key to rounding is determining which significant digits to keep and which to let go based on your needs.
In our example with (9.55)^3, after calculating the power, you might get a lengthy decimal number. However, since the original number has three significant digits, you would round your final answer to three significant digits as well. Here's a simple way to do it: look at the fourth digit (if there is one); if this digit is 5 or more, increase the third digit by one, and discard the rest. If it's less than 5, keep the third digit as is and discard the rest. As demonstrated in our step by step solution, if a calculator result is 871.287, you round it to 871, maintaining three significant digits.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small in a compact form. It is especially useful in science and engineering to easily handle and communicate very precise measurements. The format is M x 10n, where M is the number with only a single non-zero digit to the left of the decimal, and n is the power to which 10 is raised.
For instance, if you calculate (9.55)^3 and get 871, this can also be represented in scientific notation as 8.71 x 102, where 8.71 contains all the significant digits and 10 is raised to the power of 2 to place the decimal correctly for the original number's magnitude. Using scientific notation is a powerful way to keep the precision intact while making the number easier to read and work with. Whether you're dealing with astronomical distances or microscopic lengths, mastering scientific notation will unlock a new level of understanding and accuracy in your calculations.