Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\frac{p^{-4} q^{2}}{p^{2} q^{-3}}\)

Step-by-Step Solution

Answer

Expression simplifies to \( \frac{q^5}{p^6} \).

Step 1: Apply the Quotient of Powers Property

Use the Quotient of Powers Property which says \( \frac{a^m}{a^n} = a^{m-n} \) to the expression: \( \frac{p^{-4}}{p^2} \) becomes \( p^{-4-2} \) or \( p^{-6} \). Similarly, \( \frac{q^2}{q^{-3}} \) becomes \( q^{2-(-3)} \) or \( q^{2+3} \).

Step 2: Simplify the Exponents

Simplify the exponents obtained from Step 1. We have \( p^{-6} \) and \( q^5 \). Thus, the expression becomes \( p^{-6} q^5 \).

Step 3: Convert Negative Exponents to Positive Exponents

To write the final answer with positive exponents, use the property that \( a^{-m} = \frac{1}{a^m} \). This converts \( p^{-6} \) into \( \frac{1}{p^6} \). So, the expression \( p^{-6} q^5 \) becomes \( \frac{q^5}{p^6} \).

Key Concepts

Quotient of Powers

When simplifying expressions that contain exponents, one common technique to use is the "quotient of powers" rule. This rule is essential when you're dividing like bases that have exponents. It states that for any base \(a\), and any integers \(m\) and \(n\), \( \frac{a^m}{a^n} = a^{m-n} \). This means you keep the base the same and simply subtract the exponents.

• If you have \( \frac{p^{-4}}{p^2} \), using the quotient of powers rule gives \( p^{-4-2} = p^{-6} \).
• For \( \frac{q^{2}}{q^{-3}} \), it simplifies to \( q^{2-(-3)} = q^{5} \), because subtracting a negative is like adding a positive.

By simplifying each part using this rule, you come closer to the simplest form of your expression.

Negative Exponents

Negative exponents can be tricky because they essentially represent reciprocal actions. The general rule to remember is that \( a^{-m} = \frac{1}{a^m} \). This means when you see a negative exponent, you can think of it as "flipping" the base to the denominator and making the exponent positive.

• For instance, \( p^{-6} \) can be rewritten as \( \frac{1}{p^6} \). This allows you to convert what looks like a more complex part of an expression into something easier to manage.

Whenever you simplify an expression, converting all negative exponents to positive is crucial, especially when asked to provide final answers that follow this rule.

Positive Exponents

Positive exponents are straightforward to handle; they simply indicate that you multiply the base by itself as many times as the exponent denotes.

• For example, \( q^5 \) means \( q \times q \times q \times q \times q \).

Positive exponents are the preferred format for final answers because they provide a clear and standard way to express powers, avoiding the additional complexity that negative exponents introduce.Finally, after converting all negative exponents to positive, like we did for \( p^{-6} \) and maintaining the positive format for \( q^5 \), you're left with a clean and simplified expression: \( \frac{q^5}{p^6} \), which clearly uses only positive exponents.