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Chapter 10: Problem 28

Use the exponential key of a calculator to find an approximation to the nearest thousandth. $$ 13^{1.8} $$

Short Answer

Expert verified
13^{1.8} \approx 94.111

Step by step solution

01

- Enter the Base

Input the base number, which is 13, on your calculator.
02

- Access the Exponential Function

Locate and press the exponential function key, often labeled as '^' or 'y^x'. This key allows you to raise the base to a specific power.
03

- Enter the Exponent

Input the exponent, which is 1.8, using the calculator's number keys.
04

- Compute the Result

Press the '=' or 'Enter' key to calculate the exponentiation. The calculator will display the calculated value.
05

- Round to the Nearest Thousandth

Review the displayed result and round it to the nearest thousandth. This means keeping three decimal places and rounding as needed.

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Key Concepts

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

Calculator Functions
When solving problems involving exponential functions, a calculator can be an invaluable tool. Calculators have specific buttons and functions designed to handle various mathematical operations efficiently. For exponential calculations, look for the key labeled '^' or 'y^x'. This key allows you to raise a base number to the power of an exponent.
To use this function:
  • First, enter the base number using the number keys.
  • Next, press the exponential function key.
  • Finally, enter the exponent and press '=' or 'Enter' to compute the result.
This sequence ensures that your calculator performs the calculations correctly, rendering exact or approximate results depending on the capability of your calculator.
Exponents
Exponents denote how many times a number, known as the base, is multiplied by itself. For example, in the expression \(13^{1.8}\), 13 is the base, and 1.8 is the exponent. This expression means you're raising 13 to the power of 1.8, which is different from simple multiplication.
Exponents can be:
  • Whole numbers: Positive whole numbers result in the base number being multiplied by itself multiple times, e.g., \(2^3 = 2 \times 2 \times 2 = 8\).
  • Fractional: An exponent like 1.8 means taking the base to a fractional power, which is more complex but manageable using calculators.
  • Negative: This results in the reciprocal of the base raised to the corresponding positive exponent, e.g., \(2^{-3} = 1/2^3 = 1/8\).
Understanding how exponents work is crucial for the correct application of exponential functions in various mathematical problems.
Rounding Numbers
Rounding is a mathematical technique used to simplify numbers while retaining their essential value. When you have a long decimal, like the result from an exponential calculation, rounding makes it easier to work with. To round to the nearest thousandth:
  • Identify the fourth decimal place (the number immediately after the thousandth place).
  • If this digit is 5 or greater, increase the thousandth place by 1.
  • If it is less than 5, keep the thousandth place as it is.
For instance, if your calculator gives the result of \(13^{1.8}\) as 153.326279, you would round it to 153.326 because the digit after the thousandth place is 2, which is less than 5. This makes your answer more concise without significantly losing precision.

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Most popular questions from this chapter

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$ \log _{10} 2=0.3010 \quad \text { and } \quad \log _{10} 9=0.9542 $$ Evaluate each logarithm by applying the appropriate rule or rules from this section. $$ \log _{10} 36 $$

Use the change-of-base rule (with either common or natural logarithms) to find logarithm to four decimal places. \(\log _{\pi} 10\)

Find the hydronium ion concentration of the substance with the given pH. Human gastric contents, 2.0

Write each exponential statement in logarithmic form. $$ 10^{4}=10,000 $$

For \(1981-2003,\) the number of billion cubic feet of natural gas gross withdrawals from crude oil wells in the United States can be approximated by the function defined by $$ f(x)=3800+585 \log _{2} x $$ where \(x=1\) corresponds to \(1981, x=2\) to \(1982,\) and so on. (Source: Energy Information Administration.) Use this function to approximate, to the nearest unit, the number of cubic feet withdrawn in each of the following years. (a) 1982 (b) 1988 (c) 1996
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