Problem 53 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. (Lesson 8.2) $$ (-6 c)^{-4} $$

Step-by-Step Solution

Answer

The simplified form of expression is \(1/(-6c)^4\).

Step 1: Identify the base and the exponent

Here the base is \(-6c\) and the exponent is \(-4\). The negative exponent is what we want to convert.

Step 2: Apply the negative exponent rule

We know that for any nonzero number a and any integer n, \(a^{-n} = 1/a^{n}\). So we can apply this rule to our expression.

Step 3: Simplify the expression

Replacing the base and exponent in the previous rule, we have \((-6c)^{-4} = 1/(-6c)^4\).

Key Concepts

Negative Exponents

When dealing with exponents, a negative exponent can seem a bit daunting at first, but it's quite manageable once you understand the rule. A negative exponent indicates that the base should be taken as a reciprocal. In other words, instead of multiplying the base by itself a negative number of times—which isn't possible—we flip it upside down. This may sound confusing, but let's break it down.

• The rule is: for any non-zero number \(a\) and an integer exponent \(n\), \(a^{-n} = \frac{1}{a^n}\).
• This means if you see \((-6c)^{-4}\), it translates to \(\frac{1}{(-6c)^4}\).
• The negative sign in the exponent simply tells us to take the reciprocal and make the exponent positive.

Negative exponents are a shortcut to representing division more compactly. Once you switch to a positive exponent through this reciprocal method, it becomes easier to evaluate or simplify the expression.

Base and Exponent

In mathematics, when we talk about expressions involving powers, two key components are the base and the exponent. The base is the number you multiply by itself, while the exponent tells you how many times to multiply the base.

• For example, in the expression \((-6c)^{-4}\), the base is \(-6c\), and the exponent is \(-4\).
• The base can be any number or variable, and on its own it has no impact until an exponent is applied.
• The exponent signifies the operation: whether to multiply the base or involve reciprocal if negative.

This combination of a base and exponent is what allows for concise expression of large computations. Understanding how these components relate is crucial as it lays the groundwork for mastering more complex algebraic expressions.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra and it involves making an expression as simple as possible without changing its value. This often involves rewriting parts of the expression using mathematical rules, like those for exponents.

• Take the expression \((-6c)^{-4}\). By applying the negative exponent rule, you rewrite it as \(\frac{1}{(-6c)^4}\).
• Now, you work with the expression \((-6c)^4\) by evaluating it as \(-6c\cdot -6c\cdot -6c\cdot -6c\).
• Breaking it down and computing \((-6)^4\cdot c^4 = 1296c^4\) gives us a simplified form \(\frac{1}{1296c^4}\).

Simplification requires understanding the rules governing numbers and expressions, such as negative exponents or multiplication properties. The goal is clarity, making complex expressions easier to work with or solve.