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Chapter 8: Problem 1

Evaluate (a) \(\mathrm{e}^{1.6},(\mathrm{~b}) \mathrm{e}^{-1.6},(\mathrm{c}) \frac{1}{\mathrm{e}^{1.6}}\).

Short Answer

Expert verified
(a) The approximate value of \(e^{1.6}\) is 4.953. (b) The approximate value of \(e^{-1.6}\) is 0.202. (c) The approximate value of \(\frac{1}{\mathrm{e}^{1.6}}\) is 0.202.

Step by step solution

01

Evaluate \(e^{1.6}\)

To find the value of \(\mathrm{e}^{1.6}\), use a calculator or any reliable computation tool to perform the following calculation: \(\mathrm{e^{1.6}} \approx 4.953\). Therefore, \(\mathrm{e}^{1.6} \approx 4.953\).
02

Evaluate \(e^{-1.6}\)

Recall that any number raised to a negative exponent is equivalent to finding the reciprocal of that number raised to the corresponding positive exponent. So, \(\mathrm{e}^{-1.6} = \frac{1}{\mathrm{e}^{1.6}}\). Using the value of \(\mathrm{e}^{1.6}\) approximated in Step 1, compute the reciprocal: \(\frac{1}{4.953} \approx 0.202\). Therefore, \(\mathrm{e}^{-1.6} \approx 0.202\).
03

Evaluate \(\frac{1}{\mathrm{e}^{1.6}}\)

This expression is equivalent to the one computed in Step 2. Thus, \(\frac{1}{\mathrm{e}^{1.6}} \approx 0.202\). The results for the given expressions are: (a) \(\mathrm{e}^{1.6} \approx 4.953\) (b) \(\mathrm{e}^{-1.6} \approx 0.202\) (c) \(\frac{1}{\mathrm{e}^{1.6}} \approx 0.202\)

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Most popular questions from this chapter

Simplify as far as possible: (a) \(\mathrm{e}^{2 t} \mathrm{e}^{3 t}\) (b) \(3 \mathrm{e}^{2 t} \mathrm{e}^{-t}\) (c) \(\frac{\left(e^{x}\right)^{2}}{\mathrm{e}^{2 \mathrm{x}}}\) (d) \(\sqrt{\frac{e^{4 x}}{9}}\)

Solve (a) \(10^{x}=7\) (b) \(10^{x}=70\) (c) \(10^{x}=17\) (d) \(10^{\mathrm{x}}=0.7000\)

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Evaluate (a) \(\sinh 4.7\), (b) \(\cosh (-1.6)\), (c) tanh \(1.2\).
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