Problem 69 - 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. $$\frac{x^{4} y^{3}}{x^{7} y^{5}}$$

Step-by-Step Solution

Answer

\( \frac{x^4 y^3}{x^7 y^5} = \frac{1}{x^3 y^2} \).

Step 1: Apply the Division Property of Exponents

We start by applying the property for dividing powers of the same base: \( \frac{a^n}{a^m} = a^{n-m} \). This can be applied separately to each base in the fraction. For \( x \), we have \( \frac{x^4}{x^7} = x^{4-7} = x^{-3} \). For \( y \), it is \( \frac{y^3}{y^5} = y^{3-5} = y^{-2} \). Thus, \( \frac{x^{4} y^{3}}{x^{7} y^{5}} = x^{-3} y^{-2} \).

Step 2: Rewrite with Positive Exponents

To express each term with positive exponents, recall that \( a^{-n} = \frac{1}{a^n} \). Employ this property to convert: \( x^{-3} = \frac{1}{x^3} \) and \( y^{-2} = \frac{1}{y^2} \). Therefore, \( x^{-3}y^{-2} = \frac{1}{x^3} \cdot \frac{1}{y^2} = \frac{1}{x^3y^2} \).

Key Concepts

Positive Exponents

Positive exponents are a way to express multiplying a number by itself repeatedly. When you see an expression like \(x^n\), it means that \(x\) is being multiplied by itself \(n\) times. Using positive exponents is a simple and clean way to represent repeated multiplication, making calculations easier and more organized.

When working through algebra problems, it is essential to express your final answer with positive exponents whenever possible. This convention is standard as it keeps the expressions neat and easy to understand. If given a negative exponent during your calculations, remember to transform it into a positive exponent by using the reciprocal property: \(a^{-n} = \frac{1}{a^n}\). This conversion helps in simplifying the expression for better clarity and helps in understanding the relationship between multiplication and division in terms of exponents.

Division Property of Exponents

The division property of exponents is a core concept in algebra that facilitates the simplification of expressions involving powers. This property states that when you are dividing two expressions with the same base, you can subtract the exponents: \( \frac{a^n}{a^m} = a^{n-m} \). This property is handy when trying to simplify complex fractions with power terms.

For example, in our exercise \(\frac{x^4 y^3}{x^7 y^5}\), we use this property separately for both \(x\) and \(y\):

• For \(x\), \( \frac{x^4}{x^7} = x^{4-7} = x^{-3} \)
• For \(y\), \( \frac{y^3}{y^5} = y^{3-5} = y^{-2} \)

This step is crucial for reducing the expression to a simpler form before any further manipulation, such as converting negative exponents to positive ones.

Fraction Reduction

Fraction reduction is the process of simplifying a fraction to its simplest form. When reducing fractions involving variables and exponents, ensure that all expressions are fully simplified by applying the division property of exponents effectively. Once you have applied the division rule, as seen in simplifying \(\frac{x^4 y^3}{x^7 y^5}\) to \(x^{-3} y^{-2}\), the next step involves rewriting the expression with positive exponents for clarity.

To do this, employ the rule \(a^{-n} = \frac{1}{a^n}\). For our simplified expression:

• Convert \(x^{-3}\) to \(\frac{1}{x^3}\)
• Convert \(y^{-2}\) to \(\frac{1}{y^2}\)

Thus, the reduced and transformed fraction becomes \(\frac{1}{x^3 y^2}\), demonstrating an approach to simplify expressions while adhering to algebraic conventions.