/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Evaluate each expression to four... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 7

Evaluate each expression to four decimal places using a calculator. $$2.1^{1 / 3}$$

Short Answer

Expert verified
The cube root of 2.1 to four decimal places is 1.2784.

Step by step solution

01

Identify the operation

The symbol '^' indicates the operation of exponentiation. In this case, '1/3' is the exponent which means we are going to perform the cube root operation on 2.1.
02

Evaluate the cube root

Using a calculator, you will find that the cube root of 2.1 up to four decimal places is 1.2784.
03

Check the result

Ensure that the result is given to four decimal places as specified in the problem.

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Key Concepts

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

Cube Root
Understanding the cube root is essential when dealing with expressions like \(2.1^{1/3}\). The cube root of a number is a special value that, when multiplied by itself three times, will equal the original number. For example, if you have a number \(x\), finding the cube root means solving for \(y\) such that \(y^3 = x\).In the context of exponentiation, expressing the cube root as \(x^{1/3}\) uses fractional exponents. This method is powerful for calculator evaluations, simplifying operations that involve roots and powers. The cube root represented in this form makes it easier to input into a calculator, ready to perform a direct computation.
Calculator usage
Using a calculator effectively can significantly simplify the process of evaluating exponential expressions, especially when dealing with fractional exponents like \(1/3\). Most calculators have a specific button for exponentiation, often marked with a symbol like \(^\) or \(x^y\). For precise results, it's important to enter the values correctly.Here are some simple steps to use a scientific calculator for exponentiation:
  • Turn on your calculator and clear any previous calculations to start fresh.
  • Enter the base number \(2.1\) using the numeric keypad.
  • Press the exponentiation button, which might be marked as \(^\), \(x^y\), or similar.
  • Enter \(1/3\), or use parentheses to directly specify it as \((1/3)\) if necessary.
  • Press the equal sign to compute the result; the calculator will display the cube root of 2.1.
Using these steps will help ensure accuracy, especially when working with complicated expressions requiring precise calculations. Remember, practice with your calculator specific model to get familiar with its functions.
Decimal places
When asked to express a number to a certain number of decimal places, precision is important. Decimal places indicate how many digits should appear after the decimal point. For example, the number 1.2784 is expressed to four decimal places, where "2784" are the digits following the decimal. In situations like the original exercise asking for four decimal places, it is vital to round all calculations accurately. If using a calculator, ensure it is set to display the required number of decimal places or manually round off the result. Here's a quick tip:
  • Find the desired decimal place, in this case, the fourth position after the decimal point.
  • Look at the digit immediately following your desired decimal place.
  • If this digit is 5 or greater, round the last digit up. If it鈥檚 less than 5, leave the last digit as is.

By maintaining this level of precision, calculations stay consistent and reliable, as expected in mathematical tasks.

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Most popular questions from this chapter

Applications In this set of exercises, you will use inverse functions to study real-world problems. When you buy products at a store, the Universal Product Code (UPC) is scanned and the price is output by a computer. The price is a function of the UPC. Why? Does this function have an inverse? Why or why not?

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln x=(x-2)^{2}$$

Find the interest rate \(r\) if the interest on the initial deposit is compounded continuously and no withdrawals or further deposits are made. Initial amount: $$ 8500 ;\( Amount in 5 years: $$ 10,000\)

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 4000 ; r=5.75 \%\)

The spread of a disease can be modcled by a logistic function. For example, in carly 2003 there was an outbreak of an illness called SARS (Severe Acute Respiratory Syndrome) in many parts of the world. The following table gives the total momber of cases in Canada for the wecks following March 20,2003 (Source: World Health Organization) (Note: The total number of cases dropped from 149 to 140 between weeks 3 and 4 because some of the cases thought to be SARS were reclassified as other discases.) $$\begin{array}{|c|c|}\hline\text { Weeks since } & \\\\\text { March } 20,2003 & \text { Total Cases } \\\0 & 9 \\\1 & 62 \\\2 & 132 \\\3 & 149 \\\4 & 140 \\\5 & 216 \\\6 & 245 \\\7 & 252 \\\8 & 250 \\\\\hline\end{array}$$ (a) Explain why a logistic function would suit this data well. (b) Make a scatter plot of the data and find the logistic function of the form \(f(x)=\frac{\epsilon}{1+a \varepsilon^{-1}}\) that best fits the data. (c) What docs \(c\) signify in your model? (d) The World Health Organization declared in July 2003 that SARS no longer posed a threat in Canada. By analyring this data, explain why that would be so.
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