Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\((2.6)^{1.3}\)

Step-by-Step Solution

Answer

The solution to the expression \((2.6)^{1.3}\) rounded to three decimal places is 3.546.

Step 1: Evaluating the Expression

First, input the expression \((2.6)^{1.3}\) into your calculator. This will yield a result. Since calculators vary in their operation, make sure you are entering the expression correctly.

Step 2: Rounding the Result

Now, after obtaining the result, it's time to round it to the nearest three decimal places. To do this, look at the fourth decimal place. If it is 5 or more, increase the third decimal number by one, if it's less than 5, leave the third decimal number as it is.

Step 3: Final Answer

The rounded-off value of the result is the final answer to the problem.

Key Concepts

Rounding Decimal Places

Understanding how to round decimal places is crucial in ensuring precision in mathematical computations, especially when dealing with values that extend beyond our needed level of accuracy.
When rounding to a specific number of decimal places, the key is to focus on the digit right after the last desired decimal place. For example, if you are asked to round to three decimal places, you look at the fourth decimal digit.

• If this digit is 5 or greater, you increase the third decimal by one.
• If it is less than 5, you leave the third decimal as is.

This process is known as 'rounding up' or 'rounding down' respectively. An important aspect to note is the concept of 'significant figures' which plays a role in the accuracy of scientific measurements but is separate from simply rounding off decimal places.

Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, representing repeated multiplication of a number by itself. An expression like \((2.6)^{1.3}\) is an example of an exponential expression where 2.6 is the base and 1.3 is the exponent.
The calculation of such expressions can be non-trivial without a calculator, especially when the exponent is a non-integer. In cases where the exponent is a fraction or a decimal, the expression represents a root. For instance, \((2.6)^{1.3}\) can also be thought of as \(2.6\) raised to the power of 1 and 3/10. This makes the expression even more complex and emphasizes the importance of a calculator in efficiently solving these problems.

Using Calculators in Mathematics

Calculators are indispensable tools in modern mathematics education and practice. They save time and reduce the human error factor when performing complex calculations. However, their usage requires understanding the correct input methods and operation sequences. Each calculator may have a different way of handling operations, especially with more complex expressions like exponentials.
It's vital to become familiar with your calculator's instructions to avoid common errors. For instance, in scientific calculators, there is often a specific button for entering exponents. Furthermore, calculators can perform a wide range of functions beyond basic arithmetic, such as trigonometric functions, logarithms, and statistical calculations, which are essential for higher-level math coursework.