Problem 35 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(5 x^{-3}\)

Step-by-Step Solution

Answer

\(\frac{5}{x^3}\)

Step 1: Understand Negative Exponents

A negative exponent indicates that the base is on the wrong side of a fraction line. To convert a term with a negative exponent to one with a positive exponent, take the reciprocal of the base. For example, if you have a term like \(x^{-n}\), it is equivalent to \(\frac{1}{x^n}\).

Step 2: Apply the Rule for Negative Exponents

For the expression \(5x^{-3}\), apply the rule for negative exponents. The variable \(x\) with a negative exponent becomes \(\frac{1}{x^3}\). Thus, the expression \(5x^{-3}\) is transformed to \(5 \cdot \frac{1}{x^3}\).

Step 3: Simplify the Expression

Simplifying \(5 \cdot \frac{1}{x^3}\), we get \(\frac{5}{x^3}\) which is the expression without any negative exponents. Keep the coefficient 5 in the numerator since it is positive and part of the expression.

Key Concepts

Understanding Negative Exponents

Negative exponents can appear confusing at first, but they follow a simple rule. They indicate that the base should be taken to the opposite side of a fraction. Here’s how it works:
When you see something like \(x^{-n}\), think of it as needing a "flip" over the fraction line. It transforms to \(\frac{1}{x^n}\).
Let's break it down:

• \(x^{-n}\) means you take the reciprocal — the "flip" — of \(x^n\).
• Practically, every term with a negative exponent is actually saying, "Put me in the denominator with a positive exponent."

This understanding is essential to simplifying expressions where negative exponents are present. It’s like turning a "minus" into a "plus" but in a fractional context.

The Concept of a Reciprocal

A reciprocal is simply one over a number. Imagine the number as a simple fraction: its reciprocal is what it becomes when you "flip" the fraction. For instance:

• The reciprocal of 2 is \(\frac{1}{2}\).
• For a fraction like \(\frac{3}{4}\), its reciprocal is \(\frac{4}{3}\).

The principle of reciprocal is crucial when dealing with negative exponents, as they require the conversion of terms into their reciprocals to "flip" them from a numerator to a denominator. This concept allows you to rewrite expressions without negative exponents, ultimately making them more straightforward and easier to work with.

Working with Positive Exponents

Positive exponents tell you how many times to multiply the base by itself. If you have \(x^3\), you're multiplying \(x\) three times: \(x \cdot x \cdot x\).
This is a straightforward form compared to negative exponents, which require flipping. Here’s why maintaining positive exponents is useful:

• Easier to compute: You just multiply the base the specified number of times.
• Improves readability: No fractions are involved, making expressions less complex.
• Enhances clarity: Both scientific and non-scientific communities prefer expressions without negatives for ease of communication.

By converting negative exponents to positive, you simplify the mathematical expression, leaving you with a cleaner, easier-to-understand result.