Problem 25 - Elementary and Intermediate Algebra

Question

Express using positive exponents and simplify, if possible. \(b^{-5}\)

Step-by-Step Solution

Answer

The expression \( b^{-5} \) simplified with positive exponents is \( \frac{1}{b^5} \).

Step 1: Understand Negative Exponents

Negative exponents indicate that we need to take the reciprocal of the base number raised to the opposite positive exponent. For example, if we have an expression like: \( x^{-n} \), it is equivalent to \( \frac{1}{x^n} \).

Step 2: Apply the Rule to the Given Expression

Given the expression \( b^{-5} \), apply the negative exponent rule. It becomes \( \frac{1}{b^5} \) because the negative exponent suggests taking the reciprocal and changing the sign of the exponent from negative to positive.

Step 3: Simplification Check

Check if the expression \( \frac{1}{b^5} \) can be simplified further. In this case, \( \frac{1}{b^5} \) is already in its simplest form with positive exponents, so no further simplification is needed.

Key Concepts

Exponent Rules

Understanding exponent rules is crucial for handling various mathematical expressions efficiently. An exponent indicates how many times a base number is multiplied by itself. For example, in the expression \( b^3 \), \(b\) is the base and \(3\) is the exponent, signifying that \(b\) is multiplied by itself three times (\(b \times b \times b\)).
Exponent rules provide a systematic way to manipulate expressions involving powers of numbers. A key rule to remember is the negative exponent rule. This rule states that a negative exponent indicates a reciprocal. Thus, \( x^{-n} \) is equivalent to \( \frac{1}{x^n} \).
These rules help streamline calculations and are essential tools in algebra and beyond.

Simplifying Expressions

Simplifying expressions is about reducing them to their most basic form while retaining their original value. This often involves applying exponent rules and dealing with various operations such as multiplication or division.
To simplify an expression like \( b^{-5} \), we use exponent rules to express it as \( \frac{1}{b^5} \). Here, the negative exponent is converted to a positive one by taking the reciprocal of the base with the exponent.
The goal is to make the expression easier to work with, aiding further calculations or analyses. Once you understand these methods, simplifying expressions becomes a much more manageable task.

Reciprocal of a Base

The concept of the reciprocal is another important foundation in mathematical expressions. The reciprocal of a number is simply 1 divided by that number. This is particularly important when working with negative exponents.
For any base \(b\), if you have \( b^{-n} \), it can be transformed to \( \frac{1}{b^n} \). The negative exponent indicates that you should flip the base, hence using its reciprocal with a now positive exponent.
Understanding reciprocals allows for a smoother handling of expressions like fractions and makes calculations involving powers more straightforward. Mastering the reciprocal concept is part of strengthening overall mathematical problem-solving skills.