Problem 69 - Algebra and Trigonometry

Question

Use a calculator with a \(y^{x}\) key or a \(A\) key to solve. A decimal approximation for \(\sqrt{3}\) is 1.7320508 . Use a calculator to find \(2^{1.7}, 2^{1.73}, 2^{1.73}, 2^{1.73215},\) and \(2^{17300508} .\) Now find \(2^{\sqrt{3}}\). What do you observe?

Step-by-Step Solution

Answer

In carrying out these calculations, one observation is that, the closer the exponent is to the decimal approximation of \(\sqrt{3}\), the closer the result is to \(2^{\sqrt{3}}\). Thus, this exercise illustrates the impact of accurately using the value of \(\sqrt{3}\) on the final result.

Step 1: Calculate values with decimal approximations

Using a calculator, compute the values of \(2^{1.7}, 2^{1.73}, 2^{1.73}, 2^{1.73215},\) and \(2^{17300508}\).

Step 2: Calculate actual value

Now, compute the value of \(2^{\sqrt{3}}\) using the same calculator.

Step 3: Compare values

After carrying out these calculations, compare the values obtained. Observe how the result of \(2^{1.7}, 2^{1.73}, 2^{1.73}, 2^{1.73215},\) and \(2^{17300508}\) approaches the result of \(2^{\sqrt{3}}\) as the exponent approaches the decimal approximation of \(\sqrt{3}\).