Problem 56 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. (Lesson 8.2) $$ \frac{1}{(-7 m)^{-3}} $$

Step-by-Step Solution

Answer

The expression \(1/(-7m)^{-3}\) simplifies to \(-343m^3\).

Step 1: Identification of the Negative Exponent Rule

First, recognize that -3 is the exponent in the expression. For any nonzero number a, the principle \(a^{-n} = 1/a^n\) applies, allowing us to get rid of any negative exponent.

Step 2: Application of the Negative Exponent Rule

Next, apply the negative exponent rule. The original expression is \(1/(-7m)^{-3}\). So according to the rule, this becomes \(1 * (-7m)^3\), which simplifies to \(-7m * -7m * -7m\). The negative exponent has now been converted to a positive exponent.

Step 3: Simplify the Expression

Finally, simplify the expression. Calculate \(-7m * -7m * -7m\) to get \( -343m^3\). This is the final result.

Key Concepts

Exponent Rules

Understanding exponent rules is fundamental in algebra, especially when dealing with complex expressions. These rules are shortcuts that help us manipulate expressions involving exponents more efficiently.

One of the essential exponent rules is the negative exponent rule, which states that for any nonzero base 'a' and a positive integer 'n', we have \(a^{-n} = \frac{1}{a^n}\). This rule tells us that a negative exponent indicates the reciprocal of the base raised to the opposite positive power. It's a useful tool to transform negative exponents into positive ones, thus simplifying the expression.

Why do we even use negative exponents?

Negative exponents are not just a mathematical curiosity; they reveal the inverse relationship between exponential growth and decay, and they are crucial when dealing with diminutive scales or rates of decrease in various fields such as physics and finance.

When simplifying expressions, applying the negative exponent rule early on can make the process smoother and help prevent mistakes.

Simplifying Expressions

Simplifying expressions is a process to alter a complex algebraic expression into a simpler or more manageable form without changing its value. The goal is to make the expression as clear and concise as possible.

A step-by-step approach is often employed when simplifying expressions, which entails applying mathematical operations and exponent rules orderly. By following a systematic procedure, we ensure that we reduce the expression effectively and avoid errors. It's not just about making an expression look 'nicer'; it's a critical skill for solving equations, as simpler forms can be more easily manipulated or solved.

Importance in Real-World Applications

Simplified expressions can represent real-world phenomena more directly and are essential for computational efficiency in various technological applications.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific mathematical relationship or quantity. They are the building blocks for more complex algebraic equations and can be as simple as \( x + 1 \) or as intricate as \( -\frac{1}{(-7 m)^{-3}} \) in the given exercise.

Understanding algebraic expressions involves recognizing the components, like coefficients, terms, exponents, and bases, and knowing how they interact according to algebraic principles. When we face an algebraic expression with a negative exponent, it's not just about knowing the rules, but also about understanding the concept behind them; it helps to see the bigger picture and to better understand algebra's descriptive power.

Application in Various Fields

Algebraic expressions are used in virtually every domain of science and engineering, making them a necessary skill for students across many disciplines. They are used to create models that simulate real-world situations and to solve problems that would otherwise be complex or impossible to decode.