Problem 67 - 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 \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=\left(\frac{1}{3}\right)^{1-2 t}$$

Step-by-Step Solution

Answer

The function is \( f(t) = \frac{1}{3} \cdot 9^t \).

Step 1: Identify the Form

The goal is to rewrite the function in the form \( f(t) = k \cdot a^t \). We start with the given \( f(t) = \left(\frac{1}{3}\right)^{1-2t} \).

Step 2: Apply Exponent Distribution

Use the property \( a^{x+y} = a^x \cdot a^y \), and express the exponent as a sum: \( 1 - 2t = 1 + (-2t) \).

Step 3: Separate the Exponent

Apply the exponentiation rule: \( \left(\frac{1}{3}\right)^{1 + (-2t)} = \left(\frac{1}{3}\right)^1 \cdot \left(\frac{1}{3}\right)^{-2t} \).

Step 4: Simplify the Base Power

Calculate \( \left(\frac{1}{3}\right)^1 = \frac{1}{3} \). So, the expression becomes \( \frac{1}{3} \cdot \left(\frac{1}{3}\right)^{-2t} \).

Step 5: Simplify Using Power Rule

Rewrite \( \left(\frac{1}{3}\right)^{-2t} \) using the property of exponents: \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n \). Thus, \( \left(\frac{1}{3}\right)^{-2t} = 3^{2t} \).

Step 6: Combine Final Expression

Substitute back, \( f(t) = \frac{1}{3} \cdot 3^{2t} \). This is in the form \( k \cdot a^t \), where \( k = \frac{1}{3} \) and \( a = 3^2 = 9 \).

Step 7: Final Function Form

Express \( f(t) \) as \( f(t) = \frac{1}{3} \cdot 9^t \).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential when working with exponential expressions. One key property is the **product of powers** property, which states that when you multiply like bases, you add their exponents:

• \( a^{x+y} = a^x \cdot a^y \)

Applying this property allows us to break down complex expressions into simpler parts. For instance, in our exercise, we decomposed the exponent \( 1-2t \) into the sum \( 1 + (-2t) \). This expression can then be separated into two distinct parts using the product of powers rule.
Another important aspect is understanding how to handle negative exponents. The property \( a^{-x} = \frac{1}{a^x} \) helps convert negative exponents into reciprocals, facilitating simplification. For instance, in the problem, \( \left(\frac{1}{3}\right)^{-2t} \) was converted to \( 3^{2t} \), enabling further simplification.
Recognizing these properties allows students to manipulate and rewrite exponential expressions effectively, making it easier to find functional forms like \( f(t) = k \cdot a^t \). The efficiencies gained through these exponent rules simplify potentially complex algebraic manipulations into straightforward arithmetic.

Exponentiation Rules

The rules of exponentiation provide a toolkit for simplifying expressions involving powers. They include:

• **Power of a Power:** \((a^m)^n = a^{m \cdot n}\)
• **Power of a Product:** \((ab)^m = a^m \cdot b^m\)
• **Power of a Quotient:** \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\)

Each of these rules supports the process of transforming expressions into a simpler or more useful form. In the example exercise, the power of a quotient property was particularly useful.
By addressing the expression \( \left(\frac{1}{3}\right)^{-2t} \), the transformation to \( 3^{2t} \) was based on the rule \( \left(\frac{1}{3}\right)^{-n} = \left(\frac{3}{1}\right)^n \). This transformation illustrates how complex exponential expressions can be redefined using exponent rules to fit desired forms like \( f(t) = k \cdot a^t \).
Familiarity with these rules can streamline solving exponential problems, boost confidence in manipulating algebraic expressions, and improve students' mathematical fluency.

Function Simplification

Function simplification involves rewriting functions in a more condensed or useful form, often making them easier to analyze or apply. To achieve this, various mathematical skills are employed, such as recognizing patterns, using rules of arithmetic, and manipulating algebraic expressions.
In our example, the process involved displaying \( f(t) = \frac{1}{3} \cdot 9^t \) as a much simpler form than the original function \( f(t) = \left(\frac{1}{3}\right)^{1-2t} \).
By identifying constants like \( k = \frac{1}{3} \) and using exponentials effectively, the function was redefined in a more standard structure. This makes it clear how the function behaves as \( t \) changes, allowing for more straightforward graphing or computational work.
The transformation demonstrates the importance of applying algebraic techniques, as a complex or intimidating function can be converted into its simplest form. This simplification not only makes the function more understandable but also more accessible for further mathematical operations or problem-solving.