Problem 31 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Base. $$(-2.84)^{3}$$

Step-by-Step Solution

Answer

-22.9

Step 1: Understand the concept of significant digits

There are three significant digits in the base -2.84, so the calculated answer must also be rounded to three significant digits.

Step 2: Calculate the cube of the negative base

To evaluate the expression \[ (-2.84)^3 \], calculate the cube of 2.84 and then apply the negative sign. Cube of 2.84 is 2.84 * 2.84 * 2.84.

Step 3: Round to significant digits

The result from step 2 should then be rounded to retain three significant digits, as the base had three significant digits.

Key Concepts

Negative Base Exponentiation

Understanding negative base exponentiation is crucial when you encounter expressions like \( (-2.84)^3 \). It involves calculating the power of a negative number. The key lies in meticulously applying the negative sign. For any non-zero real number 'a' raised to an odd exponent 'n', the result will also be negative, just like the base. In contrast, if 'n' is even, the result is positive.

When computing \( (-a)^n \), first calculate \( a^n \), and then apply the negative sign accordingly. For example, to evaluate \( (-2.84)^3 \), you first find \( (2.84)^3 \) which is \( 2.84 \times 2.84 \times 2.84 \), and finally apply the negative sign to get the result.

Rounding Numbers

Rounding numbers is a method to reduce the digits in a number while keeping its value similar to the original number. When you round a number, you adjust it to the nearest value with fewer significant digits, based on certain rules. For instance, if you have 5 or more as the first digit after the desired number of significant digits, you round up. Otherwise, you round down.

For our expression from the exercise \( (-2.84)^3 \), the result should be rounded to three significant digits since the base, -2.84, consists of three significant digits. Remember, significant digits include all non-zero digits, zeros between non-zero digits, and trailing zeros where there is a decimal place.

Evaluation of Expressions

The evaluation of expressions involves performing the operations according to the rules of arithmetic and algebra. Consistent use of order of operations is essential. When dealing with negative bases and exponents, remember to handle the sign and magnitude separately.

Following the order of operations, for our example \( (-2.84)^3 \), you'd start by dealing with the exponentiation. The magnitude \( (2.84)^3 \) is calculated first followed by applying the negative sign. You combine these steps with an understanding of significant digits to achieve the correct level of precision in the final answer. The goal is for the student to understand not just how to perform the calculations, but also the underlying concepts that ensure the correctness of the process.