Problem 33 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Exponent. $$1^{-3}$$

Step-by-Step Solution

Answer

The expression \(1^{-3}\) evaluates to 1, retaining the proper number of significant digits.

Step 1: Understanding Negative Exponents

Negative exponents denote the reciprocal of the base raised to the absolute value of the said exponent. For any non-zero number 'a' and a negative exponent '-n', the expression can be written as: \(a^{-n} = \frac{1}{a^{n}}\).

Step 2: Apply Negative Exponent Rule

Using the rule of negative exponents, \(1^{-3} = \frac{1}{1^{3}}\). Since any nonzero number to the power of any number is itself, \(1^{3} = 1\), the expression simplifies to \(\frac{1}{1}\).

Step 3: Simplify the Expression

The fraction \(\frac{1}{1}\) simplifies to 1, because any number divided by itself equals 1. Since the base is 1 and the exponent is -3, the result maintains all significant digits, which in this case is just 1.

Key Concepts

Significant Digits

Understanding the concept of significant digits—also known as significant figures—is crucial for precisely communicating the precision of a measurement or calculation. Significant digits include all the numbers that carry meaning contributing to its precision. This includes all non-zero digits, zeros between non-zero digits, and any trailing zeros in the decimal portion. When dealing with exponents, the number of significant digits in the base is preserved after simplification, no matter how large or small the exponent value is.

For example, consider the expression from our exercise \(1^{-3}\). The base number, 1, contains only one significant digit. After the application of the exponent, the result is still 1, which has one significant digit. It is important in scientific notation that we respect the significant digits of the initial measurement throughout our calculations, as adding or losing digits can lead to inaccuracies in the representation of a number.

Reciprocal of the Base

In mathematics, the reciprocal of a number refers to the value which, when multiplied by the original number, equals 1. In simpler terms, the reciprocal of any non-zero number \(a\) is \(1/a\). When dealing with negative exponents, understanding the reciprocal becomes essential.

To illustrate, if you're given the expression \(a^{-n}\), this can be interpreted as 1 divided by \(a\) raised to the positive exponent \(n\), or \(1/a^{n}\). For example, in the expression \(1^{-3}\) from the exercise, applying this definition, we translate the negative exponent to the reciprocal of 1 raised to the 3rd power, yielding \(1/1^3\) which simplifies to \(1/1\), confirming that the reciprocal of 1 is unsurprisingly, still 1.

Simplifying Exponents

The process of simplifying exponents involves reducing expressions to their simplest form. With negative exponents, simplification typically includes converting them to their reciprocal forms and then applying the properties of exponents.

As seen in the exercise, \(1^{-3}\) simplifies to \(1/1^3\). Since \(1^3\) (1 raised to any power) equals 1, the expression simplifies down to \(1/1\), or simply 1. Regardless of the operation, whether it's raising to a power, multiplying, or taking a reciprocal, the expression simplifies to a form that is easier to comprehend and use. Recognizing that an operation with the number 1 as the base remains the number 1 is a helpful shortcut in working with exponents.