Problem 126 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(b^{-4}\right)^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \((b^{-4})^{-1}\) using only positive exponents. Answer: \(b^{4}\)

Step 1: Apply the Power Rule

First, we will apply the power rule, which states: \((a^{m})^{n} = a^{mn}\). Apply this to the given expression: $$ (\left(b^{-4}\right)^{-1} = b^{(-4)(-1)} $$

Step 2: Simplify the expression further

Now, we can simplify the expression further by multiplying the exponents: $$ b^{(-4)(-1)} = b^{4} $$ Thus, the simplified expression with only positive exponents is \(b^{4}\).

Key Concepts

The Power Rule

When studying algebra, understanding how to handle exponents is crucial. The power rule is a fundamental principle that makes working with exponents simpler. It states that when raising an exponent to another power, you multiply the exponents. For instance, when given \texttt{\((a^m)^n\)}, the power rule allows us to combine these exponents by multiplying to get \texttt{\(a^{mn}\)}. This rule streamlines the simplification of expressions involving exponents, especially when dealing with more complex algebraic equations.

Exponentiation

Exponentiation is the mathematical operation where a number, known as the base, is multiplied by itself a certain number of times, indicated by the exponent. For example, \texttt{\(2^3\)} means 2 multiplied by itself 3 times, resulting in 8. An important aspect of exponentiation is dealing with negative exponents. A negative exponent, such as \texttt{\(b^{-4}\)}, signifies taking the reciprocal of the base raised to the positive exponent, thus \texttt{\(b^{-4} = \frac{1}{b^4}\)}. The process of exponentiation is pivotal in various fields, including physics, engineering, and finance, due to its ability to express growth, decay, and compound processes.

Simplifying Expressions

Simplifying expressions is a technique used in algebra to make expressions easier to understand and work with. The goal is to condense an expression to its simplest form while maintaining its original value. This process often involves applying exponent rules, such as the power rule, to eliminate negative exponents and combining like terms. Simplifying an expression may also include factoring, distributing, and canceling out terms. When expressions are simplified, they're more manageable for performing operations such as addition, subtraction, multiplication, and division.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (such as \texttt{b}), and arithmetic operations like addition, subtraction, multiplication, and division. These expressions do not contain an equals sign, unlike equations. Algebraic expressions can represent real-world quantities and relationships, making them essential tools in solving problems in mathematics and related disciplines. An expression can be as simple as a single term, like \texttt{\(3x\)}, or as complex as a polynomial with many terms. The key to working with algebraic expressions is understanding the properties and rules of algebra, which guide how to manipulate these expressions systematically.