Problem 66 - A Graphical Approach to College Algebra

Question

Use properties of exponents to write each function in the form \(f(t)=k a^{\prime},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=3^{2 t+3}$$

Step-by-Step Solution

Answer

\( f(t) = 27 \cdot 9^t \), where \( k = 27 \) and \( a = 9 \).

Step 1: Identify the Expression

The given function is \( f(t) = 3^{2t+3} \). We need to rewrite this using exponent properties.

Step 2: Apply the Exponent Addition Property

Recall that the exponent property \( a^{x+y} = a^{x} imes a^{y} \) allows us to separate the exponents. Thus, we can write \( 3^{2t+3} = 3^{2t} imes 3^3 \).

Step 3: Simplify the Expression

Recognizing \( 3^3 \) as a constant, we compute it to find \( 3^3 = 27 \). Thus, our function becomes \( f(t) = 27 imes 3^{2t} \).

Step 4: Write in the Form f(t)=ka^b

Based on the previous simplification, the function is rewritten in the required form: \( f(t) = 27 \cdot (3^2)^t \). Here \( k = 27 \) and \( a = 3^2 = 9 \).

Key Concepts

Exponent Rules

Exponent rules are essential tools in algebra for managing and simplifying expressions involving powers. These rules help us to manipulate expressions so that they become easier to work with. One key rule discussed in this exercise is the **Exponent Addition Property**, which states that \( a^{x+y} = a^{x} \cdot a^{y} \). This property allows us to break down complex exponential expressions into simpler parts.

Let's look at how this rule applies:

• Given an expression like \( 3^{2t+3} \), the property helps us split it into \( 3^{2t} \cdot 3^3 \).
• This is useful because it isolates the constant part \( 3^3 \) and the variable part \( 3^{2t} \).

Understanding these rules is crucial for efficiently solving algebra problems that involve exponents. Always remember to check if you can apply an exponent rule to simplify your work.

Exponential Functions

Exponential functions are expressions where a constant base is raised to a variable exponent. They often have the form \( f(t) = a^t \), where \( a \) is a constant base and \( t \) is the variable exponent. These functions are powerful because they model real-world scenarios involving growth or decay.

• In our case, the function is \( f(t) = 3^{2t+3} \), which combines a base of 3 raised to a power that is both linear and shifted by +3.
• Once simplified using exponent rules, it becomes \( f(t) = 27 \cdot 3^{2t} \), showcasing how exponential functions can be expressed in various forms.

Exponential functions grow or decline rapidly, depending on the base's value. They are widely used in fields like compound interest calculations, population models, and radioactive decay.

Simplifying Expressions

Simplifying expressions is crucial in making complex algebraic expressions more manageable. It involves using mathematical properties to reduce expressions to their simplest form. In this exercise, simplifying \( f(t) = 3^{2t+3} \) involves recognizing constant factors and applying them appropriately.

• The initial step uses the exponent rule to split the expression: \( 3^{2t+3} = 3^{2t} \cdot 3^3 \).
• Next, compute the constant \( 3^3 \), which equals 27, and express the function with this constant: \( f(t) = 27 \cdot 3^{2t} \).
• This form is simpler because it identifies \( k = 27 \) as a constant factor, making it easier to analyze and identify patterns.

Simplifying expressions is a key skill in mathematics that helps keep equations tidy and solutions straightforward, ensuring they are easier to understand and apply in various contexts.