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Chapter 5: Problem 12

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\left(\frac{1}{3}\right)^{\sqrt{6}}$$

Short Answer

Expert verified
\( \left(\frac{1}{3}\right)^{\sqrt{6}} \approx 0.1464057 \).

Step by step solution

01

Understand the Expression

The expression we need to evaluate is \( \left(\frac{1}{3}\right)^{\sqrt{6}} \). This means we are raising \( \frac{1}{3} \) to the power of the square root of 6.
02

Simplify the Power

Calculate \( \sqrt{6} \). Using a calculator, \( \sqrt{6} \approx 2.4494897 \).
03

Compute the Power

Now, raise \( \frac{1}{3} \) to the power of the approximate value of \( \sqrt{6} \). Use a calculator to find the value of \( \left(\frac{1}{3}\right)^{2.4494897} \).
04

Obtain the Approximate Value

Using the calculator, \( \left(\frac{1}{3}\right)^{2.4494897} \approx 0.1464057 \). Ensure that the calculator is set to display the maximum number of decimal places possible.

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Key Concepts

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

Calculator Usage
When dealing with complex mathematical operations like exponentiation, calculators become invaluable. Most scientific calculators can handle powers and roots efficiently.
To use your calculator for an expression like \( \left(\frac{1}{3}\right)^{\sqrt{6}} \), follow these steps:
  • First, calculate the square root part. Enter 6 and use the square root function. Your calculator should display approximately 2.4494897.
  • Next, raise \( \frac{1}{3} \) to this power. Enter \( \frac{1}{3} \) or 0.3333 (if your calculator requires a decimal form), press the power button, and enter 2.4494897.
  • Ensure your calculator is set to provide the maximum number of decimal places possible to retain accuracy.
Understanding how to correctly input these values is crucial for getting the right answer. Always double-check your entries to avoid errors.
Decimal Approximation
Decimals can make or break mathematical calculations, especially when accuracy is key. Decimal approximation is a method of presenting numbers in a simpler form while maintaining an acceptable degree of precision.
When you compute \( \left(\frac{1}{3}\right)^{2.4494897} \), the exact result may not be easily expressible, so calculators give a decimal approximation. In this case, the result is approximately 0.1464057.
Here are some tips on handling decimals:
  • Understand that more decimal places mean more accuracy, so use them wisely in calculations.
  • Be aware of your calculator settings to ensure maximum decimal place display, which helps in keeping computations precise.
  • Use decimals to compare or estimate where exact figures are not necessary but maintain as many places as needed for precision.
Approximating decimals helps simplify complex numbers and keeps your work manageable.
Square Root Computation
Square root computation is an essential part of mathematics, and it's needed when dealing with expressions like \( \left(\frac{1}{3}\right)^{\sqrt{6}} \). To compute the square root of 6 accurately, you rely on calculators for exactness and ease.
Here's a straightforward way to think about it:
  • The square root of a number is a value that, when multiplied by itself, gives the original number.
  • For \( \sqrt{6} \), you find a number which when squared equals 6. Calculators quickly compute this, showing approximately 2.4494897.
  • Knowing how to find square roots helps in solving more complex problems involving powers and roots.
Understanding how to compute and use square roots allows you to handle expressions with confidence and accuracy. It's a key skill in many areas of math and science.

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Most popular questions from this chapter

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{9} 12$$

Newton's law of cooling says that the rate at which an object cools is proportional to the difference \(C\) in temperature between the object and the environment around it. The temperature \(f(t)\) of the object at time t in appropriate units after being introduced into an environment with a constant temperature \(T_{0}\) is $$f(t)=T_{0}+C e^{-k t}$$ where \(C\) and \(k\) are constants. Use this result. A pot of coffee with a temperature of \(100^{\circ} \mathrm{C}\) is set down in a room with a temperature of \(20^{\circ} \mathrm{C}\). The coffee cools to \(60^{\circ} \mathrm{C}\) after 1 hour. (a) Write an equation to model the data. (b) Estimate the temperature after a half hour. (c) About how long will it take for the coffee to cool to \(50^{\circ} \mathrm{C} ?\) Support your answer graphically.

Escherichia coli is a strain of bacteria that occurs naturally in many organisms. Under certain conditions, the number of bacteria present in a colony is approximated by $$A(t)=A_{0} e^{0.023 t}$$ where \(t\) is in minutes. If \(A_{0}=2,400,000,\) find the number of bacteria at each time. Round to the nearest hundred thousand. (a) 5 minutes (b) 10 minutes (c) 60 minutes

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Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=|x-6|$$
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