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Chapter 9: Problem 11

$$ f(x)=2^{5 x} 3^{4 x^{2}} $$

Short Answer

Expert verified
The function is f(x) = 2^{5x} 3^{4x^2}, combining exponential growths of bases 2 and 3.

Step by step solution

01

Understand the given function

The given function is f(x) = 2^{5x} 3^{4x^2}. This is an exponential function that combines powers of 2 and 3.
02

Identify necessary properties of exponents

Recall the properties of exponents: (a^m)(a^n) = a^{m+n} and (a^m)^n = a^{mn}. These properties will help in simplifying the given expression if needed.
03

Simplify or rewrite the expression (if needed)

In this particular case, no simplification is explicitly necessary for the given form. The function can be analyzed in terms of its base exponential components: 2^{5x} and 3^{4x^2}.

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Key Concepts

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

Properties of Exponents
To fully understand the function \(f(x) = 2^{5x} \, 3^{4x^2}\), it is essential to grasp the properties of exponents. Exponents are a shorthand way to express repeated multiplication. For instance, \(2^3 = 2 \times 2 \times 2\).

Some key properties of exponents include:

  • Product of Powers Property: When multiplying like bases, you add the exponents: \(a^m \, a^n = a^{m+n}\).
  • Power of a Power Property: When raising an exponent to another exponent, you multiply the exponents: \((a^m)^n = a^{mn}\).
  • Power of a Product Property: When raising a product to an exponent: \((ab)^n = a^n \, b^n\).
Understanding these properties helps in simplifying and analyzing functions that involve exponents.

For our exercise, the function \(f(x)\) makes use of both multiplication and raising to power properties, while dealing with the bases 2 and 3.
Function Analysis
Function analysis is essential for understanding the behavior and characteristics of a function over its domain.

For the function \(f(x) = 2^{5x} \, 3^{4x^2}\), it is an exponential function because it involves variables in the exponent.

  • Growth Rate: The function grows very rapidly as \(x\) increases because exponential functions increase at an exponential rate.
  • Base Contribution: Each base (2 and 3) contributes to the overall growth of the function. Specifically, \(2^{5x}\) grows linearly in its exponent, while \(3^{4x^2}\) grows quadratically in its exponent, meaning it will grow even faster.
By breaking the function down into these components, \(2^{5x}\) and \(3^{4x^2}\), we see that the growth is a combination of linear exponential and quadratic exponential behaviors, leading to very rapid increases in value.
Simplification Techniques
While the function \(f(x) = 2^{5x} \, 3^{4x^2}\) may not require immediate simplification, understanding simplification techniques can be helpful in more complex scenarios.

Here are some common techniques to simplify exponential expressions:

  • Factoring: Combine like terms wherever possible. For example, if you have \(2^x \, 2^{2x}\), you can combine them to \(2^{3x}\).
  • Using Properties of Exponents: Apply properties such as the product of powers, power of a power, and others to rewrite expressions in simpler forms.
  • Common Bases: When dealing with different bases, rewriting each component using the same base may help. For example, \(a^m \, b^n\) might be rewritten where possible using a common base.
For your given function, although no simplification is overtly required, consistent practice with these techniques will ensure a deeper understanding and an easier approach to more complex problems later on.

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Most popular questions from this chapter

A tank contains 100 gal of fresh water and brine containing \(2 \mathrm{lb}\) of salt per gallon flows into the tank at the rate of 3 gal/min. If the mixture, kept uniform by stirring, flows out at the same rate, how many pounds of salt are there in the tank at the end of 30 min?

$$ f(x)=x^{e^{x}} $$

\(y^{2} e^{2 x}+x y^{3}=1\)

Sugar decomposes in water at a rate proportional to the amount still unchanged. If there were \(50 \mathrm{lb}\) of sugar present initially and at the end of \(5 \mathrm{hr}\) this is reduced to \(20 \mathrm{lb}\), how long will it take until \(90 \%\) of the sugar is decomposed?

\(\int_{0}^{1} e^{2} d x\)
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