Problem 30 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Base. $$(-2.34)^{5}$$

Step-by-Step Solution

Answer

-64.6

Step 1: Understanding the problem

We need to evaluate the expression \( (-2.34)^5 \) and ensure that our answer retains the proper number of significant digits. In this problem, there are three significant digits in the base number (2.34). Since the base is negative and raised to an odd power, the result should also be negative.

Step 2: Calculating the power

Calculate \( (-2.34)^5 \) by multiplying \( -2.34 \) by itself four more times. As this involves repeated multiplication of a negative number, the power's sign will depend on the odd/even nature of the exponent; in this case, 5 is odd, so the result will be negative.

Step 3: Retaining significant digits

Once the calculation is complete, round the result to three significant digits to match the number of significant digits in the original base number.

Key Concepts

Significant Digits

The concept of significant digits, also known as significant figures, plays a critical role in the precision of mathematical and scientific calculations. Significant digits are the digits in a number that carry meaningful contributions to its measurement resolution. They include all non-zero digits, zeroes between non-zero digits, and trailing zeros in the decimal portion of a number. For instance, in the number 2.34, there are three significant digits: the '2', the '3', and the '4'.

When performing calculations, it's important to maintain the correct number of significant digits to ensure that the precision is not overstated. In the exercise given, we have the base number -2.34 with three significant digits. The rule is that after performing mathematical operations, the final answer should be expressed with the same number of significant digits as the original number with the least significant digits. This prevents over-speculation on the precision of computed results.

Understanding how to determine significant digits and apply them throughout calculations is essential for accuracy in measurements, especially when dealing with scientific data where precision matters.

Negative Base Exponentiation

When it comes to negative base exponentiation, it's crucial to be aware of the base number's sign and the power to which it is raised. Raising a negative number to a power involves multiplying the number by itself a specific number of times. However, the result's sign is determined by the exponent's nature - positive or negative, even or odd. For negative bases:

• If the exponent is even, the result will be positive because a negative number multiplied by itself an even number of times cancels out the negatives.
• If the exponent is odd, the result will be negative since there will be one negative number left without a pair to cancel it out.

For the expression \( (-2.34)^5 \) given in our exercise, since 5 is an odd number, we multiply -2.34 by itself four additional times, resulting in a negative value. This negative outcome is pertinent even before considering significant digits, and it's a crucial step in accurately solving such mathematical problems.

Mathematical Calculations

In the realm of mathematical calculations, precision and methodology are key to arriving at the correct results. Fundamental operations such as addition, subtraction, multiplication, and division are building blocks that require careful attention to detail, especially when dealing with significant digits and negative bases.

For our example, the operation involves exponentiation, a form of repeated multiplication. This requires careful sequential arithmetic to ensure that each step is performed correctly. The process begins with understanding the number's components (base and exponent), following through with the multiplication carefully, and finally, correctly rounding the result to maintain the appropriate level of significant digits.

It is also important to recognize that different mathematical operations have specific rules for significant digits. For multiplication and division, the number of significant digits in the final product or quotient should match the number with the fewest significant digits used in the calculation. This is why in our exercise, after evaluating \( (-2.34)^5 \) and obtaining the numerical result, it is then necessary to round off that result to three significant digits, respecting the precision of the initial data.