Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$4^{-2 t}=0.0625$$

Step-by-Step Solution

Answer

The solution to the exponential equation \(4^{-2 t}=0.0625\) is \(t = 1\).

Step 1: Identify the Given Equation

Here, the equation to be solved is \(4^{-2 t}=0.0625\). The goal is to find the value of the variable \(t\).

Step 2: Simplify the Equation

Notice that the right side of equation, 0.0625, can also be written as \( \frac{1}{4^2} \) or \(4^{-2}\). This same base simplification produces the equation \(4^{-2t} = 4^{-2}\).

Step 3: Equate the Exponents

When an exponential equation \^{a^{m} = a^{n}}\ is simplified to have the same base, the only possibility is that the exponents must be equal. This allows us to set \(-2t = -2\).

Step 4: Solve for the Variable

By dividing both sides by -2, we calculate that \(t = -2/-2 = 1\). Therefore, the solution to the equation \(4^{-2t} = 0.0625\) is \(t = 1\).

Step 5: Verification

Substituting \(t = 1\) into the original equation, \(4^{-2(1)} = 0.0625\), checks out since \(4^{-2} = 0.0625\). Also, the result can be verified graphically by showing that the function \(y = 4^{-2t}\) intersects the line \(y = 0.0625\) at \(t = 1\).

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is used to express equations and relationships in a symbolic form. When working with equations, the primary goal is to find the value of the variable that makes the equation true. In the context of the exercise, we are solving for the variable \( t \) in the equation \( 4^{-2t} = 0.0625 \). Algebraic techniques such as simplifying expressions, equating powers, and solving for unknowns allow us to manipulate the equation to isolate the variable.
Furthermore, algebra involves operations like addition, subtraction, multiplication, and division to solve for a variable. These operations are crucial when working with exponential equations. We simplify the equation by recognizing that both sides can be rewritten using the same base, a common algebraic technique that makes solving equations more manageable.

Exponents

Exponents signify repeated multiplication of a number by itself. In algebra, they are written as a superscript. For example, in \( 4^3 \), the exponent is 3, meaning 4 is multiplied by itself three times (\( 4 \times 4 \times 4 \)). When solving exponential equations like \( 4^{-2t} = 0.0625 \), understanding exponents is essential.
Negative exponents indicate a reciprocal. Thus, \( 4^{-2} \) means \( \frac{1}{4^2} \), which equals 0.0625. Recognizing this equivalence allows us to rewrite the equation in a simpler form, \( 4^{-2t} = 4^{-2} \). Simplifying the exponents where the base is the same enables directly equating the exponents themselves, simplifying further algebraic operations.
Exponents are foundational to many mathematical concepts, making them crucial knowledge for solving exponential equations.

Graphing Utility

A graphing utility is a powerful tool in mathematics for visually representing functions and verifying solutions. It helps plot complex equations on a graph, providing insights that algebraic manipulation alone might not reveal. In this exercise, we use a graphing utility to confirm our solution after solving \( 4^{-2t} = 0.0625 \) algebraically.
By graphing the function \( y = 4^{-2t} \) and the horizontal line \( y = 0.0625 \), we can observe where these two graphs intersect. The intersection point represents the value of \( t \) that satisfies the equation. In our exercise, the graphical verification showed that the intersection occurs at \( t = 1 \), confirming our algebraic solution.
Graphing utilities can vary from handheld calculators to sophisticated software programs, helping visualize concepts and verify algebraic solutions.

Verification Methods

Verification methods are essential to ensure that a solution to an equation is correct. After obtaining an algebraic solution, verifying it introduces a layer of accuracy and credibility to mathematical work. For exponential equations, there are two common verification methods: substitution and graphing.
First, substituting the solved value back into the original equation checks if it satisfies the equation. In our exercise, inputting \( t = 1 \) into \( 4^{-2(1)} = 0.0625 \) confirms the solution, as it holds true mathematically.
Second, graphing provides a visual method for verification. By examining the graphs of the function and a horizontal line representing the equation's right side, we check that they intersect at the solution point. This dual approach of substitution and graphing ensures robust verification of the solution.