Problem 51 - College Algebra

Question

A decimal approximation for \(\sqrt{3}\) is \(1.7320508 .\) Use a calculator to find \(2^{1.7}, 2^{1.73}, 2^{1.732}, 2^{1.73205},\) and \(2^{1.7320508}\) Now find \(2^{\sqrt{3}} .\) What do you observe?

Step-by-Step Solution

Answer

You will observe that as the exponent gets closer and closer to the actual value of \(\sqrt{3}\), the calculated result significantly gets closer to the value of \(2^\sqrt{3}\). This shows that as you use more precise approximations, the result gets closer to the actual value.

Step 1: Calculate Powers of 2

Use a calculator to find the decimal values of \(2^{1.7}, 2^{1.73}, 2^{1.732}, 2^{1.73205},\) and \(2^{1.7320508}\). These are approximations for \(2^\sqrt{3}\) since the numbers \(1.7, 1.73, 1.732, 1.73205, 1.7320508\) are approximations for \(\sqrt{3}\).

Step 2: Calculate the Actual Value

Now calculate \(2^\sqrt{3}\) using a calculator. Here, do not use an approximation for \(\sqrt{3}\) , but the actual value. This is the actual result with which you'll compare your previous results.

Step 3: Observing Patterns

Now that you have all the results, look at the series of results for all approximations of \(2^1.7320508\) . Do you observe any patterns? Specifically, watch how the results change as the approximation for \(\sqrt{3}\) gets more and more precise. Compare these results to the actual value for \(2^\sqrt{3}\)

Key Concepts

Calculator Usage in Algebra

Algebra, a fundamental branch of mathematics, often requires the manipulation of numbers and variables to solve equations and understand patterns. When tackling more complex calculations such as those involving exponents and roots, a calculator becomes an invaluable tool. However, using a calculator effectively is a skill in itself.

Firstly, it’s crucial to understand the order of operations that your calculator follows - most employ the standard PEMDAS/BODMAS rules. Knowing how to input expressions correctly can prevent errors. When dealing with exponents and roots, use the specific buttons designed for these functions. For example, to find \(2^{1.7}\), you would press the buttons for 2, exponent, 1.7, and then equals (

• 2 ^ 1.7 =

) on a scientific calculator.

Keep an eye out for any rounding that the calculator may automatically do, as this can impact the accuracy of your results. Also, be aware that different calculators may have varying precisions. In educational settings, using calculators helps students understand the concept of limits and approximations when exact values cannot be easily computed. Through step-by-step exercises such as approximating \(2^{\sqrt{3}}\), students get hands-on experience in seeing how increased precision in an approximation (like adding more decimal places to \(\sqrt{3}\)) results in a value that gets successively closer to the true value.

Exponents and Powers

Exponents, also known as powers, are a shorthand notation in mathematics for expressing repeated multiplication of a number by itself. The exponent indicates how many times the base number is to be used in a multiplication. For instance, \(2^3\) means \(2 \times 2 \times 2\), which equals 8.

In algebra, exponents are fundamental in describing growth patterns, decay processes, and many natural phenomena. When dealing with exponents, there are several important rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \( (a^m)^n = a^{m \times n}\)
• Power of a Product: \( (ab)^n = a^n \times b^n\)

In the exercise provided, \(2^{1.7}\) and the other powers of 2 are incrementally precise approximations to \(2^{\sqrt{3}}\). This exercise is a practical application of understanding exponents, demonstrating how they are used to explore mathematical concepts like irrational numbers (\(\sqrt{3}\)) and their properties within the real number system. Understanding these can help students grasp how even tiny changes in the exponent can significantly affect the overall value of a number raised to a power.

Square Roots in Algebra

The square root of a number is a value that, when multiplied by itself, gives the original number. For a positive number \(a\), the square root, denoted as \(\sqrt{a}\), is the positive solution to the equation \(x^2 = a\). Square roots are a specific case of roots in algebra, which involve finding a number that raises to a certain power to give the original number.

In algebra, square roots are particularly important when solving quadratic equations and in understanding the properties of quadratic functions. The concept of the square root bridges into the geometric interpretation as well: the square root of an area of a square gives the length of its side.

It is essential to recognize that not all numbers have a simple square root. For instance, the square root of most non-square numbers will be an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation is non-repeating and non-terminating. \(\sqrt{3}\), as seen in the original exercise, is one such irrational number. In a calculative scenario, we often use approximations of irrational roots, and as the exercise shows, the more decimal places used in the approximation, the closer the result is to the actual irrational number. Understanding square roots and their approximations allow students to solve a wide range of algebraic problems with greater precision and to appreciate the complexity of numbers beyond simple integers and fractions.