Problem 25 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Base. $$(-8.01)^{3}$$

Step-by-Step Solution

Answer

\(-512.48\)

Step 1: Understanding Exponentiation with Negative Bases

First, recognize that raising a negative base to an odd power will result in a negative answer. This is because an odd number of negative factors multiplied together will always give a negative product.

Step 2: Calculating the Cube of the Base

Calculate the cube of the absolute value of the base, ignoring the negative sign for now. Cube the value of 8.01 to get: \(8.01 \times 8.01 \times 8.01\) or \(8.01^3\).

Step 3: Applying the Negative Sign and Significant Digits

Since the base was negative and raised to the third power, which is odd, apply the negative sign to the result from Step 2. Also, ensure that the result has the correct number of significant digits. The base, 8.01, has three significant digits, so the final answer should have three significant digits as well. Rounded properly, the final answer would be \(-512.48\).

Key Concepts

Negative Base Exponentiation

When it comes to negative base exponentiation, it's essential to understand how negative numbers behave under exponent rules. If a negative base is raised to an odd exponent, the result will be negative because a negative number multiplied by itself an odd number of times will always end in a negative product. Conversely, if the exponent is even, the result will be positive since the negatives cancel out. This principle is fundamental in evaluating expressions like \( (-8.01)^{3} \), where the base is negative, and the exponent is 3, an odd number. Always begin by considering the absolute value of the base, then apply the exponent, and finally reintroduce the negative sign if needed. In the provided exercise, after calculating the cube of 8.01, the negative sign is reapplied to indicate the correct direction on the number line.

Calculating Cube of a Number

To calculate the cube of a number, which is raising a number to the power of three, simply multiply the number by itself twice. For the number 8.01, cubing it involves calculating \(8.01 \times 8.01 \times 8.01\) or \(8.01^{3}\). This operation results in the base number growing at a much faster rate compared to squaring, as it is multiplied by itself an additional time. With each multiplication, the digits can significantly change, especially with non-integers, making it crucial to handle significant digits correctly to preserve the precision of scientific calculations.

Applying Significant Digits Rules

Significant digits are a crucial concept in mathematics and scientific measurement, as they convey how precise a number is. To follow the rules of significant digits:

• Non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Leading zeros are never significant.
• Trailing zeros are significant if the number contains a decimal point.

When it comes to operations, the number of significant digits in the result is determined by the original number with the fewest significant digits. After performing calculations, it's important to round the result to the appropriate number of significant digits to retain the intended precision. In the example \( (-8.01)^{3} \), since the base 8.01 has three significant digits, the final result, after considering the rules of significant digits and rounding, is \( -512.48 \) which retains three significant digits as per the base number.