Problem 70 - 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. $$\frac{x^{5} y^{2}}{x^{6} y^{3}}$$

Step-by-Step Solution

Answer

The reduced form is \(\frac{1}{xy}\).

Step 1: Apply the Exponential Division Property

Start by applying the property \(\frac{a^{n}}{a^{m}} = a^{n-m}\) separately to both the variables \(x\) and \(y\). For \(x\), this gives \(x^{5-6}\) and for \(y\), this gives \(y^{2-3}\).

Step 2: Simplify the Exponents

Calculate the exponents from the previous step. For \(x^{5-6}\), it simplifies to \(x^{-1}\). For \(y^{2-3}\), it simplifies to \(y^{-1}\). So the expression becomes \(x^{-1}y^{-1}\).

Step 3: Convert Negative Exponents to Positive

Use the property \(a^{-n} = \frac{1}{a^{n}}\) to rewrite the expression with positive exponents. Thus, \(x^{-1} = \frac{1}{x}\) and \(y^{-1} = \frac{1}{y}\). Therefore, the combined expression becomes \(\frac{1}{x} \times \frac{1}{y} = \frac{1}{xy}\).

Key Concepts

Negative Exponents

Negative exponents might seem daunting at first, but they are quite simple to understand with a bit of practice. Imagine the whole idea is about turning multiplications into divisions, and vice versa. When you see an exponent that is negative, such as in the term \(x^{-n}\), it means you should take the reciprocal of the base raised to the positive value of the exponent. For instance, \(x^{-3} = \frac{1}{x^3}\). It's like flipping the fraction marked by the base and then applying the positive exponent to it.

Here's a useful checklist for negative exponents:

• Convert the negative exponent by writing the inverse (reciprocal) of the base.
• After flipping, raise the base to the absolute value of the exponent.
• Simplify if possible.

While transitioning from negative to positive exponents, always confirm each step is accurate—solve carefully! This is especially vital when performing operations with multiple bases and exponents in an expression.

Fraction Reduction

Reducing fractions is all about simplifying them to their simplest form, and there’s a rule that greatly aids in this task: the Exponential Division Property \(\frac{a^n}{a^m} = a^{n-m}\). This property allows you to subtract the exponent of the denominator from the exponent of the numerator as long as they have the same base.

For example, if you see \(\frac{x^5}{x^6}\), you can express it using the Exponential Division Property as \(x^{5-6} = x^{-1}\). This step effectively reduces the fraction by canceling out common bases. The tricky part is ensuring that only positive exponents remain after the reduction, which may require you to apply the negative exponent rule later on.

• Identify common bases in both numerator and denominator.
• Apply the division property by subtracting the exponents.
• Simplify until all exponents are positive.

Not only does this process eliminate unnecessary complexity, but it also paves the way for easier computations later on.

Positive Exponents

Positive exponents are the simplest to deal with. They indicate straightforward multiplication. The expression \(x^3\) simply means \(x\times x \times x\). But when dealing with an expression involving negative exponents, converting these to positive is essential for clarity and simplicity.

Imagine taking \(x^{-1}\) and changing it to its positive equivalent. This involves flipping the situation to \(\frac{1}{x^1}\) or simply \(\frac{1}{x}\). Each part of a multi-step process leads to an eventual expression that is much friendlier to work with, especially in fraction forms increased through positive exponents.

• Convert using \(a^{-n} = \frac{1}{a^n}\) for any negative powers.
• Multiply or divide as indicated in the structure of the problem.
• Simplify to ensure that the expression is expressed with only positive exponents.

Positive exponents help in maintaining consistency and accuracy within mathematical expressions, and knowing how to convert negative to positive flawlessly streamlines problem solving.