Problem 68 - Elementary Algebra

Question

The link between positive and negative exponents \(\left(a^{-n}=\frac{1}{a^{n}}\right)\) along with the property \(\frac{a^{n}}{a^{m}}=a^{n-m}\) can also be used when reducing fractions. Consider this example: $$ \frac{x^{3}}{x^{7}}=x^{3-7}=x^{-4}=\frac{1}{x^{4}} $$ Use this approach to express each fraction in reduced form. Give all answers with positive exponents only. $$\frac{x^{4}}{x^{8}}$$

Step-by-Step Solution

Answer

\( \frac{x^{4}}{x^{8}} = \frac{1}{x^{4}} \)

Step 1: Apply Exponent Subtraction Rule

Start by using the exponent rule for division: \( \frac{a^{n}}{a^{m}} = a^{n-m} \). This rule allows you to subtract the exponents of like bases when dividing. Here, apply this rule to \( \frac{x^{4}}{x^{8}} \), resulting in \( x^{4-8} = x^{-4} \).

Step 2: Convert to Positive Exponent

To express the result with positive exponents, use the rule for negative exponents: \( a^{-n} = \frac{1}{a^{n}} \). Apply this to \( x^{-4} \), converting it to \( \frac{1}{x^{4}} \) to have a positive exponent.

Key Concepts

Negative Exponents

Negative exponents might seem confusing at first, but they have a pretty straightforward logic behind them. When you encounter a negative exponent, it simply means the reciprocal of the base raised to the absolute value of the exponent. In other words, if you have something like \( a^{-n} \), this is equivalent to \( \frac{1}{a^n} \). This rule is really useful in algebra, especially when simplifying expressions.
Here are some key points about negative exponents:

• A negative exponent indicates division by the base, not multiplication.
• It helps transform large, complex expressions into simpler, more manageable fractions.

For instance, in our example \( x^{-4} \) turns into \( \frac{1}{x^4} \). This conversion makes it easier to interpret and work with expressions. Practicing this conversion will help make working with negative exponents much simpler and more intuitive.

Positive Exponents

Positive exponents tell you how many times to multiply a number, or base, by itself. They are perhaps the concept that many find the easiest to grasp. For instance, \( a^n \) simply means multiplying \( a \) by itself \( n \) times. This is a fundamental building block in algebra and pre-algebra math.
When reducing fractions with positive exponents, such as \( \frac{x^4}{x^8} \), we look for ways to subtract these positive numbers. Here, by using the exponent rule for division, you perform the subtraction (\( 4 - 8 \)), which transforms positive exponents into a single term, which often results in negative exponents.
Understanding positive exponents is crucial because they:

• Provide the basis for division rules in exponents.
• Are essential for simplifying and solving equations.

Once you have addressed negative exponents, positive exponents usually simplify to their original form, like turning \( x^{-4} \) back to a positive exponent by reversing it: \( \frac{1}{x^4} \). Remember, the goal in mathematics is often to work towards keeping expressions as simple as possible.

Fraction Reduction

Fraction reduction in algebra is made convenient through the use of exponent laws. Recognizing and simplifying components of an expression can reduce unnecessary complexity in fractions. Let's break it down:
When reducing fractions that contain exponents, one key rule is the subtraction rule: \( \frac{a^n}{a^m} = a^{n-m} \). This allows a clean reduction almost instantly by just dealing with exponent adjustments. For example, \( \frac{x^4}{x^8} \) boils down to \( x^{-4} \), thanks to subtracting \( 8 \) from \( 4 \).
Simplifying these fractions effectively:

• Reduces expressions to their simplest form.
• Prepares them for further arithmetic operations or evaluations.

A focus on converting negative results to positive, as shown by the transformation of \( x^{-4} \) to \( \frac{1}{x^4} \), prepares your expressions for further mathematical tasks. This simplification not only aids clarity but ensures accurate computations moving forward. Be it in handling equations, expressions, or any algebraic operations, fraction reduction through exponent rules is an invaluable skill.