Problem 55 - College Algebra with Modeling and Visualization

Question

Exercises \(55-64:\) Use the power rules to simplify the expression. Use positive exponents to write your answer. $$ \left(5^{-1}\right)^{3} $$

Step-by-Step Solution

Answer

\( \frac{1}{125} \)

Step 1: Apply the Power of a Power Rule

The power of a power rule states that when you raise a power to another power, you multiply the exponents: \( (a^m)^n = a^{m imes n} \). Here, apply the rule: \( (5^{-1})^3 = 5^{-1 imes 3} = 5^{-3} \).

Step 2: Rewrite with Positive Exponents

Convert the negative exponent to a positive exponent using the rule \( a^{-m} = \frac{1}{a^m} \). Therefore, \( 5^{-3} = \frac{1}{5^3} \).

Step 3: Evaluate the Expression

Calculate \( 5^3 \) which is \( 5 \times 5 \times 5 = 125 \). Substitute this into the expression: \( \frac{1}{5^3} = \frac{1}{125} \).

Key Concepts

Exponents

Exponents are incredibly useful mathematical tools that help us express repeated multiplication in a concise format. Instead of writing a long sequence of the same number multiplied by itself, we use an exponent to indicate how many times the base number is used as a factor. For instance, when we write \(5^3\), we mean \(5\times5\times5\), which results in the product 125.

In general, an expression in the form of \(a^n\) signifies that the base \(a\) is multiplied by itself \(n\) times. This simple yet powerful concept makes it easier to handle large numbers and complex calculations. Understanding how exponents work is essential as they form the foundation of more advanced algebraic rules.

Negative Exponents

Negative exponents can be a little tricky at first, but once you grasp the simple rule behind them, they become quite manageable. A negative exponent, like \(a^{-m}\), means that instead of multiplying the base \(a\), you actually take its reciprocal \(\frac{1}{a^m}\). This is a rule that turns division into multiplication by inverting or flipping the base's power.

For example, if you have \(5^{-3}\), it turns into \(\frac{1}{5^3}\). This transformation makes simplifying expressions with negative exponents straightforward. Just remember: a negative exponent tells you to take the reciprocal of the base raised to the positive exponent. This rule allows us to rewrite such expressions in an easier-to-work-with format, often necessary before performing further calculations.

Simplifying Expressions

Simplifying expressions is all about making them as easy to handle as possible without changing their value. When dealing with exponents, simplification often involves using rules like the power of a power rule or the negative exponent rule.

- **Power of a Power Rule:** When you have an exponent raised to another power, you multiply the exponents. For instance, \((5^{-1})^3\) simplifies to \(5^{-1\times3} = 5^{-3}\). This rule helps break down complex exponentiation into more manageable steps.
- **Positive Exponents:** Always try to express your final answer in terms of positive exponents. For example, convert \(5^{-3}\) to \(\frac{1}{5^3}\). This shift emphasizes clarity and simplicity.

Once the expression is fully simplified, evaluate the numerical value if needed. For instance, \(\frac{1}{5^3}\) evaluates to \(\frac{1}{125}\), providing a neat and comprehensible result. Simplifying expressions not only helps in solving algebraic problems but also enhances understanding of the mathematical processes involved.