Problem 26 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Base. $$(-1.71)^{5}$$

Step-by-Step Solution

Answer

-7.83

Step 1: Identify the Number of Significant Digits

Before computing the value, identify the number of significant digits in the base of the expression, which is '-1.71'. There are three significant digits in '-1.71'.

Step 2: Calculate the Fifth Power of the Negative Base

Compute the value of (-1.71)^{5}. Because the exponent is an odd number, the result will remain negative. Use a calculator to perform the exponentiation with the base to the fifth power.

Step 3: Round the Answer to the Correct Number of Significant Digits

After calculating the value, round the answer to the same number of significant digits as the base. Since the base has three significant digits, the answer should also be rounded to three significant digits.

Key Concepts

Negative Base Exponentiation

Negative base exponentiation is a crucial concept in mathematics, especially when dealing with polynomial and algebraic expressions. When raising a negative base to an exponent, the sign of the result is determined by the power. If the exponent is an odd number, the result retains the negative sign; if it's even, the result will be positive. In the given exercise, \( (-1.71)^5 \) is evaluated. The base is negative and since the exponent 5 is an odd number, the final answer will also be negative. It's important to use brackets around the negative base to avoid confusion with negative exponents.

Working through this involves calculating the absolute value of the base raised to the power first—here \( 1.71^5 \)—and then applying the negative sign to the result. This operation often requires a calculator, especially with non-integer bases and when high accuracy is needed. Remember to correctly interpret the negative sign in conjunction with the exponent during your calculations.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It's particularly useful in science and engineering to simplify measurements and calculations. In scientific notation, a number is written as the product of two factors: a coefficient that is greater than or equal to 1 but less than 10, and a power of 10. For example, the number 650,000 is written as \( 6.5 \times 10^5 \) in scientific notation.

When dealing with significant digits, the coefficient will reflect this precision. In calculations, it's critical that the scientific notation is used to maintain the proper number of significant digits. Scientific notation also makes it easier to perform operations like multiplication and division, as you can work with the coefficients and add or subtract the exponents respectively. Understanding how to convert numbers into and out of scientific notation is an essential skill when working with significant digits.

Rounding Numbers

Rounding numbers is a fundamental concept in mathematics, employed to reduce the lengthy digits to a manageable count while retaining the value's precision as much as possible. When rounding to a specific number of significant digits, as in our negative base exponentiation exercise, you first identify the number of significant digits in the original number, then round the final result to that same count. For \( (-1.71)^5 \) with three significant digits in the base, our rounded result will also contain exactly three significant digits.

Applying Rounding Rules

If the digit following the last significant digit is greater than or equal to 5, increase the last significant digit by one. If it's less than 5, the last significant digit remains unchanged. These rules ensure that the rounded number is as close as possible to the original number in terms of actual value. It's also important to consider the context of the problem when rounding, as rounding can impact the accuracy of the result in mathematical and real-world applications.