Problem 44 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ a^{-4} b^{-6} c^{-1} d^{4} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression with only positive exponents: \(a^{-4} b^{-6} c^{-1} d^{4}\). Answer: \(\frac{d^4}{a^4b^6c^1}\)

Step 1: Rewrite the terms with negative exponents using positive exponents

Rewrite each term with a negative exponent by applying the property \(a^{-n}=\frac{1}{a^n}\): $$ \frac{1}{a^4} \frac{1}{b^6} \frac{1}{c^1} d^4 $$

Step 2: Combine the terms

Now, we can combine the fractions: $$ \frac{d^4}{a^4b^6c^1} $$ The final expression with only positive exponents is: $$ \frac{d^4}{a^4b^6c^1} $$

Key Concepts

Negative Exponents

Understanding negative exponents is crucial in algebra and represents a key part of grasping more complex concepts. Simply put, a negative exponent tells us that we should take the reciprocal of the base raised to the absolute value of the exponent. For instance, when we encounter an expression like \( a^{-n} \), this is equivalent to \( \frac{1}{a^n} \).

It's important to note that any nonzero number raised to a negative exponent will result in a fraction. This doesn't mean the number is negative; rather, it's the exponent that's negative. For instance, \( 2^{-3} \) would be rewritten as \( \frac{1}{2^3} \) or \( \frac{1}{8} \). This rule ensures that all expressions can be expressed with positive exponents, simplifying the work with them.

Exponent Rules

Exponent rules are essential for simplifying algebraic expressions effectively. These rules make dealing with powers straightforward. One such rule is the Product of Powers rule, which states that when you multiply two powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).

Multiplying and Dividing with the Same Base

Another pair of rules is used for multiplying and dividing. The Quotient of Powers rule tells us that for the same base, you subtract the exponents when dividing: \( a^m \div a^n = a^{m-n} \). For negative exponents, this can mean making exponents more negative or converting negative exponents to positive ones.

Power of a Power

Additionally, the Power of a Power rule is applied when you raise a power to another power, multiplying the exponents: \( (a^m)^n = a^{m \times n} \). Each of these rules works hand in hand with the others to quickly and efficiently simplify expressions with exponents.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (such as \( x \) and \( y \)), and operators (such as addition, subtraction, multiplication, and division). The beauty of algebraic expressions lies in their use to represent real-world situations and their versatility in being manipulated according to algebraic rules to simplify or solve them.

For example, the expression \( 5x + 3 \) illustrates how we can combine numbers and variables. The variable can represent any number, allowing the expression to capture a relationship between quantities, like the total cost (\( 5x + 3 \)) with \( x \) representing the number of items bought and 3 representing a fixed shipping cost.

Simplifying Expressions

The process of simplifying expressions is a fundamental technique to make algebraic equations and expressions easier to work with. Simplification can involve combining like terms, using exponent rules, and performing arithmetic operations in a strategic manner. The goal is to rewrite the expression in the simplest form possible without changing its value.

In the context of exponents, to simplify an expression, you would first eliminate any negative exponents by rewriting them as positive exponents, as per the rule \( a^{-n} = \frac{1}{a^n} \). Then, you would apply other exponent rules to combine terms with the same base and simplify further. The idea is to transform the expression into one that is easier to understand and use, especially when it comes to solving equations and evaluating expressions.