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Chapter 3: Problem 7

Approximate each number using a calculator. Round your answer to three decimal places. $$e^{2.3}$$

Short Answer

Expert verified
The approximate value of \(e^{2.3}\), rounded to three decimal places, should be obtained after following these steps correctly.

Step by step solution

01

Understanding the problem

The exercise requires calculating the value of the exponential function \(e^{2.3}\). This is Euler's number, denoted as \(e\), raised to the power of 2.3.
02

Calculating the value

Using a calculator, input \(e^{2.3}\) by entering \(e\), then the exponent 2.3. Make sure to follow the correct order of operations, using parentheses if necessary. Retrieve the calculated result displayed on the calculator
03

Rounding down the result

Round the result from the calculator to three decimal places. This is done by looking at the fourth decimal place. If it is 4 or less, round down the third decimal place. If it is 5 or more, round up the third decimal place.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euler's number
Euler's number, denoted by the symbol \(e\), is a fundamental mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is crucial in calculus, especially in the study of exponential growth and decay.

When you encounter an expression like \(e^{2.3}\), you're dealing with an exponential function where Euler's number is the base and it鈥檚 being raised to a power鈥攊n this case, 2.3. It's essential to understand that \(e\) is not just a number, but it represents continuous growth and appears in many areas of mathematics, from compound interest to the shapes of curves.
Scientific calculator usage
To calculate the value of expressions involving Euler's number, like \(e^{2.3}\), a scientific calculator is an indispensable tool.

Here鈥檚 how to use one for an exponential calculation:
  • Locate the button marked \(e^x\) or \(EXP\).
  • Enter the exponent value, which is 2.3 in the given exercise.
  • Press equals to obtain the result.
It's important to follow the correct order of operations. If your expression has more components, use parentheses to signify which parts should be computed first. Scientific calculators are designed to handle complex expressions, so increase your familiarity with their functions to enhance your math-solving capabilities.
Rounding decimals
Rounding decimals is a way to simplify a number while retaining its value as closely as possible to the original. When rounding to three decimal places, you will look at the fourth decimal spot.

Here's the general rule:
  • If the fourth decimal is 5 or higher, you increase the third decimal by one (this is called rounding up).
  • If it's 4 or lower, keep the third decimal as is (rounding down).
Password doing this, ensures your answer is both accurate and easier to work with for any subsequent calculations. In practical terms, when you round \(e^{2.3}\), if the fourth decimal place is 5 or more, round the third decimal place up, otherwise leave it as it is.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can use any positive number other than 1 in the changeof-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. Normal, unpolluted rain has a pH of about 5.6. What is the hydrogen ion concentration? b. An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? c. How many times greater is the hydrogen ion concentration of the acidic rainfall in part (b) than the normal rainfall in part (a)?

Use the Leading Coefficient Test to determine the end behavior of the graph of \(f(x)=-2 x^{2}(x-3)^{2}(x+5)\) (Section 2.3, Example 2)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(3 x-2)=2$$

Describe the quotient rule for logarithms and give an example.
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