Problem 71 - 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. $$\frac{28 a^{2} b^{3}}{-7 a^{5} b^{3}}$$

Step-by-Step Solution

Answer

The fraction simplifies to \(-\frac{4}{a^3}\) with positive exponents.

Step 1: Simplify the Coefficients

Divide the numerical coefficients. In this case, divide 28 by -7, which gives -4.

Step 2: Apply the Quotient Property to 'a' Exponents

Use the property \(\frac{a^m}{a^n} = a^{m-n}\) for the \(a\) terms in the expression. We have \(\frac{a^2}{a^5} = a^{2-5} = a^{-3}\).

Step 3: Apply the Quotient Property to 'b' Exponents

The \(b\) terms are \(b^3\) in both the numerator and the denominator. Thus, \(\frac{b^3}{b^3} = b^{3-3} = b^0\). And recall that \(b^0 = 1\), so it simplifies to 1.

Step 4: Express Result with Positive Exponents

Now we have \(-4 \cdot a^{-3} \cdot 1 = -4a^{-3}\). To write this with positive exponents, we use \(a^{-n} = \frac{1}{a^n}\), resulting in \(-\frac{4}{a^3}\).

Key Concepts

Negative Exponents

Negative exponents can seem a bit confusing at first, but they're not as tricky as they appear. In algebra, a negative exponent indicates that the base has to be in the denominator of a fraction. For example, if you have \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\).
The negative sign in the exponent essentially tells you to "flip" the base. This rule allows us to reframe problems involving negative exponents into familiar fractions. Understanding this concept helps make operations involving negative exponents much more straightforward. This is essential when you're simplifying algebraic expressions because you want all exponents to be positive in your final answer.

Quotient Property of Exponents

The quotient property of exponents is a powerful tool when dealing with complex fractions involving variables raised to exponents. This property states that dividing two exponents with the same base, like \(\frac{a^m}{a^n}\), is the same as subtracting the exponents: \(a^{m-n}\).

• If \(m > n\), you'll end up with a positive exponent in the numerator.
• If \(m = n\), you will have an exponent of 0, meaning that the fraction simplifies to 1 since any number to the zero power is 1.
• If \(m < n\), the result is a negative exponent, which allows further simplification using the rules of negative exponents.

Applying the quotient property simplifies algebraic expressions and makes them more manageable.

Positive Exponents

When dealing with algebraic expressions, positive exponents are usually preferred because they are more straightforward to work with and understand. A positive exponent simply tells you how many times to multiply the base by itself.
For instance, in the last part of the exercise, we turned \(a^{-3}\) into positive form by rewriting it as \(\frac{1}{a^3}\). This step is key because having all positive exponents simplifies calculations and makes expressions cleaner. Positive forms are generally easier to interpret, making them ideal for the final form of solutions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a foundational skill in algebra that lets you present expressions in their simplest form, making it easier to solve equations. The process involves several steps:

• First, simplify the coefficients by dividing numbers just like in regular arithmetic.
• Next, apply the properties of exponents, such as the quotient property, to simplify the terms. This often involves converting negative exponents into positive ones.
• Finally, ensure all exponents are positive by placing any negative exponent terms in the denominator of a fraction.

These techniques make solving and understanding algebraic expressions more intuitive and less fraught with potential for error.