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Chapter 5: Problem 10

Simplify the expression without a calculator $$ e^{3 x} e^{1+x} $$

Short Answer

Expert verified
The simplified expression is \( e^{4x + 1} \).

Step by step solution

01

Apply the Properties of Exponents

The first step is to apply the property of exponents which states that when multiplying powers with the same base, you add their exponents. Here, both terms have the base \( e \). So, we can write: \[ e^{3x} imes e^{1+x} = e^{(3x) + (1+x)} \] Expand the exponents in the parenthesis by distributing the addition.
02

Combine Like Terms in the Exponent

In the expression \((3x) + (1 + x)\), distribute the addition and combine like terms. Calculate: \[ (3x) + (1 + x) = 3x + 1 + x = 4x + 1 \] Now substitute back into the exponent: \[ e^{3x} imes e^{1+x} = e^{4x + 1} \]
03

Write the Final Simplified Expression

Now, with the simplified exponent, you can write the entire expression. The final simplified version of the original expression is: \[ e^{4x + 1} \]

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Key Concepts

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

Base of an Exponent
The base of an exponent is such an important component when working with powers. It is essentially the number that gets multiplied by itself. In our example of simplifying the expression \( e^{3x} e^{1+x} \), the base is \( e \). Think of the base as the foundation of a building; without it, the exponential expression wouldn't have anything to "stand on."

Key points to remember about the base:
  • The base appears consistently in both terms you're multiplying, like \( e \) in this example.
  • Exponents act on the base. Thus, a consistent base like \( e \) simplifies combining different parts of an expression.
A consistent base lets you apply exponential rules more easily, saving you time and avoiding confusion.
Multiplication Rule for Exponents
When you multiply exponential terms with the same base, you add their exponents. This is called the multiplication rule for exponents. It's a powerful property that can simplify otherwise complex expressions, such as \( e^{3x} e^{1+x} \).

Here's how it works:
  • Identify the common base (e.g., \( e \) in our case).
  • Add the exponents together, keeping the base constant. For example, \( e^{3x + (1 + x)} \) becomes \( e^{4x + 1} \).
This rule is a direct result of repeated multiplication and serves as a shortcut. You save time by skipping individual multiplications and moving straight to addition of exponents. Always ensure that the bases are indeed the same before applying this rule, as it only works when the bases match.
Exponentiation Simplification
Exponentiation simplification is all about making long expressions more manageable. Simplification techniques help in combining and reducing parts of an expression to its most compact form. Our expression started as \( e^{3x} e^{1+x} \), and through simplification, we transformed it into the more straightforward \( e^{4x + 1} \).

Steps to simplify include:
  • Apply properties of exponents, such as the multiplication rule, to add exponents.
  • Incorporate basic arithmetic to combine terms within the exponent. For instance, combine \( 3x + x \) into \( 4x \).
  • Write the simplified expression with a single base-exponent structure wherever possible.
Simplification is key for clearer communication and easier calculation. It not only makes complex expressions easier to handle but also enhances understanding by keeping expressions clean and neat.

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Most popular questions from this chapter

Explain how linear and exponential functions differ. Give examples.

Hurricanes Hurricanes are some of the largest storms on earth. They are very low pressure areas with diameters of over 500 miles. The barometric air pressure in inches of mercury at a distance of \(x\) miles from the eye of a severe hurricane is modeled by the formula \(f(x)=0.48 \ln (x+1)+27\) (a) Evaluate \(f(0)\) and \(f(100)\). Interpret the results. (b) Graph \(f\) in \([0,250,50]\) by \([25,30,1] .\) Describe how air pressure changes as one moves away from the eye of the hurricane. (c) At what distance from the eye of the hurricane is the air pressure 28 inches of mercury?

The number of females working in automotive repair is increasing. The table shows the number of female ASE-certified technicians for selected years. $$\begin{array}{ccccc}\text { Year } & 1988 & 1989 & 1990 & 1991 \\ \hline \text { Total } & 556 & 614 & 654 & 737\end{array}$$ $$\begin{array}{|ccccc}\hline \text { Year } & 1992 & 1993 & 1994 & 1995 \\\\\hline \text { Total } & 849 & 1086 & 1329 & 1592\end{array}$$ (a) What type of function might model these data? (b) Use least-squares regression to find an exponential function given by \(f(x)=a b^{x}\) that models the data. Let \(x=0\) correspond to 1988 (c) Use \(f\) to estimate the number of certified female technicians in \(2005 .\) Round the result to the nearest hundred.

Graph \(f\) and state its domain. $$f(x)=\ln (-x)$$

Give an example of data that could be modeled by a logistic function and explain why.
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