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Chapter 8: Problem 3

Simplify as far as possible: (a) \(\frac{\mathrm{e}^{2 x} \mathrm{e}^{x}}{\mathrm{e}^{-3 x}}\) (b) \(\left(4 \mathrm{e}^{2}\right)\left(3 \mathrm{e}^{-x}\right)\) (c) \(\frac{2 \mathrm{e}^{x}+1}{2}+\frac{2-\mathrm{e}^{x}}{3}\) (d) \(\mathrm{e}^{4 x}-\left(\mathrm{e}^{2 x}+1\right)^{2}\)

Short Answer

Expert verified
Question: Simplify the following expressions involving exponential functions: (a) \(\frac{\mathrm{e}^{2 x} \mathrm{e}^{x}}{\mathrm{e}^{-3 x}}\) (b) \(\left(4 \mathrm{e}^{2}\right)\left(3 \mathrm{e}^{-x}\right)\) (c) \(\frac{2 \mathrm{e}^{x}+1}{2}+\frac{2-\mathrm{e}^{x}}{3}\) (d) \(\mathrm{e}^{4 x}-\left(\mathrm{e}^{2 x}+1\right)^{2}\) Answer: (a) \(\mathrm{e}^{6 x}\) (b) \(12 \mathrm{e}^{2 - x}\) (c) \(\frac{4 \mathrm{e}^{x}+7}{6}\) (d) \(-2 \mathrm{e}^{2 x}-1\)

Step by step solution

01

(a) Expression

Given expression: \(\frac{\mathrm{e}^{2 x} \mathrm{e}^{x}}{\mathrm{e}^{-3 x}}\)
02

(a) Step 1: Apply Exponent Properties

Combine exponents in the numerator using property \(a^{m} a^{n} = a^{m+n}\): \(\frac{\mathrm{e}^{2 x + x}}{\mathrm{e}^{-3 x}} \implies \frac{\mathrm{e}^{3 x}}{\mathrm{e}^{-3 x}}\)
03

(a) Step 2: Simplify the Expression

Use the property \(\frac{a^m}{a^n} = a^{m-n}\): \(\mathrm{e}^{3 x - (-3 x)} \implies \mathrm{e}^{6 x}\)
04

(b) Expression

Given expression: \(\left(4 \mathrm{e}^{2}\right)\left(3 \mathrm{e}^{-x}\right)\)
05

(b) Step 1: Distribute the Constants

Perform the multiplication of constants: \(4 \cdot 3 \mathrm{e}^2 \mathrm{e}^{-x} \implies 12 \mathrm{e}^2 \mathrm{e}^{-x}\)
06

(b) Step 2: Combine the Exponents

Use the property \(a^m a^n = a^{m+n}\): \(12 \mathrm{e}^{2 - x}\)
07

(c) Expression

Given expression: \(\frac{2 \mathrm{e}^{x}+1}{2}+\frac{2-\mathrm{e}^{x}}{3}\)
08

(c) Step 1: Find Common Denominator

The common denominator for both fractions is 6. Rewrite each fraction with the common denominator: \(\frac{3(2 \mathrm{e}^{x}+1)}{6}+\frac{2(2-\mathrm{e}^{x})}{6}\)
09

(c) Step 2: Combine the Fractions

Add the two fractions now that they have a common denominator: \(\frac{3(2 \mathrm{e}^{x}+1)+2(2-\mathrm{e}^{x})}{6}\)
10

(c) Step 3: Simplify the Expression

Distribute the constants and combine like terms in the numerator: \(\frac{6 \mathrm{e}^{x}+3+4-2 \mathrm{e}^{x}}{6} \implies \frac{4 \mathrm{e}^{x}+7}{6}\)
11

(d) Expression

Given expression: \(\mathrm{e}^{4 x}-\left(\mathrm{e}^{2 x}+1\right)^{2}\)
12

(d) Step 1: Expand the Square

Apply the formula \((a+b)^2 = a^2 + 2ab + b^2\) on the second term: \(\mathrm{e}^{4 x}-\left(\mathrm{e}^{4 x}+2 \mathrm{e}^{2 x}+1\right)\)
13

(d) Step 2: Simplify the Expression

Combine like terms and simplify: \(\mathrm{e}^{4 x}-\mathrm{e}^{4 x}-2 \mathrm{e}^{2 x}-1 \implies -2 \mathrm{e}^{2 x}-1\)

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Most popular questions from this chapter

Variables \(y\) and \(t\) are thought to be connected by a law of the form $$ y=\frac{a^{t}}{k} $$ Measurements of \(t\) and \(y\) are (a) By using appropriate graph paper find the law connecting \(y\) and \(t\). (b) Predict \(y\) when \(t=12\). (c) Predict the value of \(t\) when \(y\) first exceeds 1000

The temperature, \(T\), of a chemical reaction is given by $$ T=120 \mathrm{e}^{0.02 t} \quad t \geq 0 $$ Calculate the time needed for the temperature to (a) double its initial value, (b) treble its initial value.

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 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) \(\mathrm{e}^{4 \mathrm{x}}=90\) (b) \(10^{x / 2}-20=0\) (c) \(3 \mathrm{e}^{-x}=20\) (d) \(10^{3 \pi-6}=40\)
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