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Chapter 8: Problem 3

Simplify as far as possible: (a) \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}+\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\) (b) \(\mathrm{e}^{x}+\frac{1}{\mathrm{e}^{x}}-\mathrm{e}^{-x}\) (c) \(\frac{\mathrm{e}^{2 x}+\mathrm{e}^{x}}{\mathrm{e}^{x}}-1\) (d) \(\mathrm{e}^{3 x}\left(\mathrm{e}^{-2 x}-\mathrm{e}^{-3 \mathrm{x}}\right)+1\)

Short Answer

Expert verified
Question: Simplify the following expressions and identify if any two expressions are equal: (a) \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}+\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\) (b) \(\mathrm{e}^{x}+\frac{1}{\mathrm{e}^{x}}-\mathrm{e}^{-x}\) (c) \(\frac{\mathrm{e}^{2 x}+\mathrm{e}^{x}}{\mathrm{e}^{x}}-1\) (d) \(\mathrm{e}^{3 x}\left(\mathrm{e}^{-2 x}-\mathrm{e}^{-3 x}\right)+1\) Answer: All four expressions simplify to \(\mathrm{e}^{x}\). Therefore, they are all equal to each other.

Step by step solution

01

Combine the fractions

We can add the fractions by adding the numerators and keeping the common denominator. Result: \(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}+\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}\)
02

Simplify the numerator

The terms \(\mathrm{e}^{x}\) and \(\mathrm{e}^{-x}\) will cancel out. Result: \(\frac{2\mathrm{e}^{x}}{2}\)
03

Cancel the common factors

The factors of 2 will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\) (b) Simplify \(\mathrm{e}^{x}+\frac{1}{\mathrm{e}^{x}}-\mathrm{e}^{-x}\)
04

Express terms with the same exponent

Rewrite the second term with the same exponent as the third term: \(\frac{1}{\mathrm{e}^{x}} = \mathrm{e}^{-x}\) Result: \(\mathrm{e}^{x}+\mathrm{e}^{-x}-\mathrm{e}^{-x}\)
05

Simplify the expression

The terms \(\mathrm{e}^{-x}\) will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\) (c) Simplify \(\frac{\mathrm{e}^{2 x}+\mathrm{e}^{x}}{\mathrm{e}^{x}}-1\)
06

Factor out common exponent in numerator

Factor out \(\mathrm{e}^{x}\) in the numerator. Result: \(\frac{\mathrm{e}^{x}(\mathrm{e}^{x}+1)}{\mathrm{e}^{x}}-1\)
07

Cancel the common factors

The factors of \(\mathrm{e}^{x}\) will cancel out. Result: \(\mathrm{e}^{x}+1-1\)
08

Simplify the expression

The terms 1 will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\) (d) Simplify \(\mathrm{e}^{3 x}\left(\mathrm{e}^{-2 x}-\mathrm{e}^{-3 x}\right)+1\)
09

Distribute the exponent

Distribute the \(\mathrm{e}^{3 x}\) to both terms inside the parentheses. Result: \(\mathrm{e}^{3 x}\mathrm{e}^{-2 x}-\mathrm{e}^{3 x}\mathrm{e}^{-3 x}+1\)
10

Simplify the exponents

Using the property \(\mathrm{e}^{a}\mathrm{e}^{b}=\mathrm{e}^{a+b}\): Result: \(\mathrm{e}^{x}-\mathrm{e}^{0}+1\)
11

Simplify the expression

Since \(\mathrm{e}^{0}=1\), we combine the terms. Result: \(\mathrm{e}^{x}-1+1\)
12

Final simplified expression

The terms -1 and 1 will cancel out, and we are left with just \(\mathrm{e}^{x}\). Result: \(\mathrm{e}^{x}\)

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Most popular questions from this chapter

Simplify to a single logarithmic expression: (a) \(\ln 4 y+\ln x\) (b) \(3 \ln t^{2}-2 \ln t\) (c) \(3 \log t-\log 3 t\) (d) \(\log 2 x+\log 5 x-1\)

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

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) \(\log 2 x=1.5\) (b) \(\log (3 x+1)=2.1500\) (c) \(\log \left(x^{2}+3\right)=2.3671\) (d) \(4 \log (5 x-6)=-0.8000\)

Simplify each expression as far as possible: (a) \(\mathrm{e}^{2 x} \mathrm{e}^{7 x}\) (b) \(\left(3 \mathrm{e}^{x}\right)\left(2 \mathrm{e}^{-x}\right)\) (c) \(\mathrm{e}^{2 x}\left(\mathrm{e}^{-2 x}+\mathrm{e}^{-x}+1\right)-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) (d) \(\frac{\mathrm{e}^{-3 x}}{2 \mathrm{e}^{-x}}\)
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