Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Use properties of exponents to write each function in the form \(f(t)=k a^{t},\) 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

The function is \( f(t) = 27 \cdot 9^t \).

Step 1: Identify the Exponential Function

The given function is \( f(t) = 3^{2t + 3} \). It is important to recognize the form \( a^{x+y} \). In this function, \( a = 3 \), and the exponent is \( 2t + 3 \).

Step 2: Separate the Exponent Using Exponent Rule

Apply the exponent rule \( a^{x+y} = a^{x} \cdot a^{y} \) to separate the exponent into two terms: \( f(t) = 3^{2t} \cdot 3^3 \).

Step 3: Calculate the Constant

The term \( 3^3 \) can be calculated to find the constant \( k \). So, \( 3^3 = 27 \).

Step 4: Express the Function in the Required Form

Now that you have separated \( 3^{2t} \) and calculated \( 3^3 \), express \( f(t) \) in the form \( f(t) = k \, a^t = 27 \cdot (3^2)^t \).

Step 5: Simplify the Function

To make the expression clearer, write \( (3^2)^t \) as \( (3^2)^t = (9)^t \). Hence, \( f(t) = 27 \cdot 9^t \). This is in the form where \( k = 27 \) and \( a = 9 \).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with exponential functions. Exponents help express repeated multiplication in a compact form. For example, the expression \(a^n\) means multiplying \(a\) by itself \(n\) times. Here are key properties:

• Product of Powers: \(a^{m+n} = a^m \cdot a^n\). This property tells us that when you multiply numbers with the same base, you can add their exponents.
• Power of a Power: \((a^m)^n = a^{m \cdot n}\). When an exponent applies to a power, multiply the exponents.
• Power of a Product: \((ab)^n = a^n \cdot b^n\). If a product is raised to an exponent, each factor is raised to the exponent individually.

These rules simplify expressions and help rearrange terms in equations. In the original exercise, recognizing the form \(a^{x+y}=a^x \cdot a^y\) allows separating exponents for further simplification.

Exponentiation

Exponentiation is the process of raising a number, referred to as the base, to a power, which is the exponent. For example, in \(3^{2t+3}\), 3 is the base and \(2t+3\) is the exponent. It indicates how many times the base is used as a factor.

• Exponential Growth: As the exponent increases, the base number grows rapidly, which explains why exponential functions frequently model rapid growth or decay.
• Simplification: In complex expressions, breaking the exponents into familiar terms helps manage calculations, especially using properties like \(a^{x+y} = a^x \cdot a^y\).

When solving problems, breaking down exponents into manageable parts assists in simplifying and solving equations, as seen in the exercise where \(3^{2t+3}\) becomes \(3^{2t} \cdot 3^3\).

Exponential Form Conversion

Exponential form conversion is key to expressing functions in a simplified or specific format. Consider how the equation is rearranged and simplified in the given exercise.First, we separate the components of the exponent using properties of exponents: \(3^{2t+3} = 3^{2t} \cdot 3^3\). It helps in decomposing terms into recognizable parts to make the function easier to work with. Following this, calculate \(3^3\) which equals 27, providing the constant factor.Moreover, converting \(3^{2t}\) to \((3^2)^t\) and simplifying it to \(9^t\) allows the expression \(f(t) = 27 \cdot 9^t\) to conform to the desired form \(f(t) = k \cdot a^t\). This follows the principle of clear representation, turning a seemingly complicated function into a more manageable and interpretable format, making it easier to graph or analyze. Understanding and using exponential form conversion is essential in mathematics for effectively managing and solving real-world problems.