Evaluate (6e)^(-√3) To Three Significant Digits

In the realm of mathematics, evaluating expressions involving constants like e (the base of the natural logarithm) and irrational numbers such as $\sqrt$3} often requires the use of calculators to obtain accurate numerical approximations. In this comprehensive guide, we will delve into the process of evaluating the expression $(6e)^{-\sqrt{3}$ to three significant digits using a calculator. This exploration will not only demonstrate the practical application of calculators in mathematical problem-solving but also highlight the importance of understanding significant digits in numerical approximations.

Understanding the Expression $(6e)^{-\sqrt{3}}$

To begin, let's dissect the expression $(6e)^{-\sqrt{3}}$. This expression involves several mathematical concepts: exponents, the constant e, and the square root of 3. Understanding each component is crucial for accurate evaluation.

• The Constant e: The constant e is a fundamental mathematical constant, approximately equal to 2.71828. It is the base of the natural logarithm and appears in various mathematical contexts, including exponential growth and calculus.
• The Square Root of 3 ($\sqrt{3}$): The square root of 3 is an irrational number, approximately equal to 1.73205. It is the positive real number that, when multiplied by itself, equals 3.
• Exponents: The expression involves an exponent, specifically $-\sqrt{3}$. A negative exponent indicates a reciprocal, meaning $(6e)^{-\sqrt{3}}$ is equivalent to $\frac{1}{(6e)^{\sqrt{3}}}$.

Step-by-Step Evaluation Using a Calculator

Now, let's proceed with the step-by-step evaluation of the expression $(6e)^{-\sqrt{3}}$ using a calculator.

Step 1: Calculate $e^{\sqrt{3}}$

First, we need to calculate $e^{\sqrt{3}}$. Most scientific calculators have an $e^x$ function. Input $\sqrt{3}$ into the calculator and then use the $e^x$ function to find the value of $e^{\sqrt{3}}$.

$\sqrt{3} \approx 1.73205$

$e^{\sqrt{3}} \approx e^{1.73205} \approx 5.6522$

Step 2: Multiply by 6

Next, multiply the result from Step 1 by 6:

$6e^{\sqrt{3}} \approx 6 \times 5.6522 \approx 33.9132$

Step 3: Take the Reciprocal

Since the original expression has a negative exponent, we need to take the reciprocal of the result from Step 2:

$(6e)^{-\sqrt{3}} = \frac{1}{(6e)^{\sqrt{3}}} \approx \frac{1}{33.9132} \approx 0.029486$

Step 4: Round to Three Significant Digits

Finally, round the result to three significant digits. Significant digits are the digits in a number that carry meaning contributing to its precision. In this case, we need to round 0.029486 to three significant digits.

The first three significant digits are 2, 9, and 4. The next digit is 8, which is greater than or equal to 5, so we round up the last significant digit.

$0. 029486 \approx 0.0295$

Therefore, $(6e)^{-\sqrt{3}}$ evaluated to three significant digits is approximately 0.0295.

Significance of Significant Digits

Significant digits play a crucial role in scientific and mathematical calculations. They indicate the precision of a measurement or calculation. When rounding to a certain number of significant digits, we are essentially indicating the level of certainty we have in the result.

In the context of this problem, rounding to three significant digits means that we are confident in the first three digits of the result, but the remaining digits may not be as reliable. This is particularly important when dealing with approximations and calculations involving irrational numbers, where the exact value cannot be expressed in a finite number of digits.

Alternative Calculator Input

Many scientific calculators allow you to input the entire expression $(6e)^{-\sqrt{3}}$ directly, which can streamline the calculation process. Here's how you can do it:

1. Enter 6.
2. Multiply by e (use the $e^x$ function).
3. Raise the result to the power of $-\sqrt{3}$ (use the $x^y$ or ^ function).
4. The calculator will display the result, which you can then round to three significant digits.

This method can reduce the risk of manual errors and provide a more efficient way to evaluate complex expressions.

Common Mistakes to Avoid

When using a calculator to evaluate expressions, there are several common mistakes to avoid:

• Incorrect Order of Operations: Make sure to follow the correct order of operations (PEMDAS/BODMAS) when entering the expression into the calculator.
• Rounding Errors: Avoid rounding intermediate results, as this can lead to inaccuracies in the final answer. Round only the final result to the desired number of significant digits.
• Calculator Mode: Ensure your calculator is in the correct mode (degrees or radians) for trigonometric functions, if applicable.
• Input Errors: Double-check your input to avoid typos or incorrect entries.

Practical Applications

The ability to evaluate expressions involving constants and irrational numbers has numerous practical applications in various fields, including:

• Physics: Calculating physical quantities such as exponential decay, radioactive decay, and oscillations.
• Engineering: Designing electrical circuits, analyzing structural stability, and modeling fluid dynamics.
• Finance: Computing compound interest, calculating present and future values, and modeling investment growth.
• Statistics: Determining probabilities, analyzing statistical distributions, and conducting hypothesis tests.

Conclusion

In conclusion, evaluating the expression $(6e)^{-\sqrt{3}}$ to three significant digits using a calculator involves a series of steps, including calculating $e^{\sqrt{3}}$, multiplying by 6, taking the reciprocal, and rounding to the appropriate number of significant digits. This process highlights the importance of understanding mathematical concepts, using calculators effectively, and being mindful of significant digits in numerical approximations. By mastering these skills, you can confidently tackle a wide range of mathematical problems and applications.

The result, rounded to three significant digits, is approximately 0.0295. This detailed exploration not only provides the answer but also enhances your understanding of the underlying mathematical principles and the practical use of calculators in mathematical problem-solving.