Problem 36 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(27 m^{-3}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{27}{m^3}\).

Step 1: Identify the Negative Exponent

The negative exponent in the expression is in the term \(m^{-3}\). Recall that negative exponents indicate the reciprocal of the base raised to the positive of that exponent.

Step 2: Rewrite the Expression with Positive Exponents

To rewrite \(m^{-3}\) with a positive exponent, you take the reciprocal of \(m^{-3}\). So, \(m^{-3}\) becomes \(\frac{1}{m^3}\).

Step 3: Combine Terms

Rewrite the whole expression by substituting \(m^{-3}\) with \(\frac{1}{m^3}\). The expression becomes \(27 \cdot \frac{1}{m^3}\) or \(\frac{27}{m^3}\).

Key Concepts

Negative Exponents

Negative exponents can seem daunting at first, but they're actually quite straightforward. A negative exponent means that instead of multiplying the base by itself a certain number of times, you divide 1 by the base raised to the corresponding positive exponent.
For example, with the expression \( m^{-3} \), you have a base \( m \) with a negative exponent of -3. This means you would take the reciprocal of \( m^3 \), which results in \( \frac{1}{m^3} \). So, \( m^{-3} \) is equal to \( \frac{1}{m^3} \).
This principle is a key part of simplifying equations involving negative exponents. By converting negative exponents into their reciprocal form, you often make it easier to perform operations or further simplify the expression.

Simplifying Expressions

Simplifying expressions is about transforming a complex or cumbersome expression into a more manageable form without changing its value.
When dealing with expressions like \( 27m^{-3} \), the goal is to rewrite it in a simpler form by making all exponents positive.
First, notice the negative exponent in \( m^{-3} \)—rewrite this as \( \frac{1}{m^3} \). Now, insert this back into the original expression to get \( 27 \cdot \frac{1}{m^3} \), which simplifies down to \( \frac{27}{m^3} \).
This new expression is simpler because it no longer contains a negative exponent, making it easier to interpret and work with for further calculations or context.

Reciprocals

The concept of reciprocals is crucial when working with negative exponents.
A reciprocal is simply what you multiply a number by to get 1. In fraction terms, for any non-zero number \(a\), its reciprocal is \(\frac{1}{a}\).
Understanding reciprocals helps you handle expressions where the base is raised to a negative exponent. When you see \(m^{-3}\), instead of thinking of it directly as a negative power, think of it as \(1\) divided by \(m^3\).
This shift in perspective, recognizing that the reciprocal operation transforms the negative exponent into a positive one, is very helpful. It's a foundational concept in algebra to convert and simplify expressions effectively.