Problem 12 - Elementary and Intermediate Algebra

Question

Use a calculator to determine whether 2.5646 is an approximate solution of \(2^{2 x+1}=70\)

Step-by-Step Solution

Answer

Yes, 2.5646 is an approximate solution.

Step 1: Understand the Equation

The given equation is \(2^{2x+1}=70\). We need to determine if \(x = 2.5646\) is an approximate solution to this equation.

Step 2: Substitute x into the Equation

Replace \(x\) with 2.5646 in the equation to get \(2^{2(2.5646)+1}\). This simplifies to \(2^{5.1292}\).

Step 3: Calculate the Exponentiation

Use the calculator to compute \(2^{5.1292}\). Enter 2, raise it to the power of 5.1292 to find the result, which should be approximately 69.79.

Step 4: Compare with the Right-hand Side

Compare the calculated value, approximately 69.79, with the right side of the equation, which is 70. Since 69.79 is close to 70, \(x = 2.5646\) is an approximate solution.

Key Concepts

Approximate Solutions

In mathematics, sometimes exact solutions are not always practical or necessary, especially when dealing with real-world problems or complex equations. Here comes the concept of approximate solutions. When you determine that a certain value for a variable brings the result of a given equation very close to the expected outcome, you are identifying an approximate solution. Approximate solutions are useful when:

• The precise value is difficult or impossible to compute.
• An exact answer isn't required for practical purposes but a close estimate suffices.
• The answer provides insights that are almost as valuable as the exact number.

In the exercise at hand, the approximate solution arises from comparing a computed result with a target value. If the two values are close enough by a reasonable measure, the solution is accepted as approximate. Understanding how to derive and evaluate approximate solutions allows one to tackle equations where precise computation might either be cumbersome or unwarranted.

Calculator Use in Problem-Solving

Calculators are essential tools in modern problem-solving. They've evolved to handle complex arithmetic which includes handling powers, roots, and logarithms. Using a calculator efficiently can save time, reduce errors, and simplify complex calculations down to something manageable. When using calculators, remember to:

• Enter operations in the correct sequence to ensure accurate calculations.
• Be mindful of rounding, as this can affect the accuracy of your result.
• Familiarize yourself with specific functions your calculator offers. These can greatly aid in specific tasks like exponentiation.

In our exercise, the calculator computes the value of an exponential equation by evaluating \(2^{5.1292}\) accurately. Such calculations would be tedious to perform by hand, highlighting the importance of calculators in problem-solving. Embracing these tools allows mathematicians and students alike to focus on interpreting results rather than getting bogged down in computation.

Exponentiation

Exponentiation is a mathematical operation, written as \(a^n\), involving two numbers: the base \(a\) and the exponent (or power) \(n\). It entails multiplying the base by itself as many times as the number denoted by the exponent. This concept is a fundamental part of higher mathematics, and it appears in many real-world applications, including interest calculations, population growth models, and more. Key points about exponentiation:

• The base \(a\) is the number that gets multiplied.
• The exponent \(n\) shows how many times the base is used as a factor.
• For example, \(2^3\) is equal to \(2 \times 2 \times 2 = 8\).
• Calculating large powers manually can be cumbersome, which makes calculators invaluable.

In the problem, exponentiation of 2 to the power of 5.1292 is performed to verify if a specific \(x\) value is an approximate solution to the given equation. Mastery of exponentiation concepts and calculator use enables students to tackle similar exponential equations with confidence and precision.