Problem 61 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\frac{a^{3} b^{-2}}{a^{-2} b^{-4}}\)

Step-by-Step Solution

Answer

The simplified expression is \( a^5 b^2 \).

Step 1: Apply the Quotient Rule

Use the quotient rule of exponents, which states that \( \frac{x^m}{x^n} = x^{m-n} \), to simplify the expression. Apply this to both the \(a\) and \(b\) terms separately. \[\frac{a^3}{a^{-2}} = a^{3 - (-2)} = a^{3 + 2} = a^5\] \[\frac{b^{-2}}{b^{-4}} = b^{-2 - (-4)} = b^{-2 + 4} = b^2\] So, the expression becomes \( a^5 b^2 \).

Step 2: Simplify the Expression

Now that the expression is simplified, we have \(a^5 b^2\). Both exponents are positive numbers. There are no further simplifications necessary as there are no zero or negative integers as exponents.

Key Concepts

Quotient Rule of Exponents

The quotient rule of exponents is a handy tool when you need to simplify expressions involving powers with the same base. This rule applies when dividing two expressions that have the same base but different exponents. Using the quotient rule, \[ \frac{x^m}{x^n} = x^{m-n} \] we subtract the exponent in the denominator from the exponent in the numerator. Let's consider a practical example:

• If you have \( \frac{a^3}{a^{-2}} \), you subtract \(-2\) from \(3\), resulting in \( a^{3-(-2)} = a^{3+2} = a^5 \).
• For \( \frac{b^{-2}}{b^{-4}} \), you have \(-2\) minus \(-4\), which results in \( b^{-2-(-4)} = b^{-2+4} = b^2 \).

This process helps in reducing the complexity of the terms in a given expression, making it easier to handle and understand.

Positive Exponents

When dealing with exponents, it is crucial to remember that positive exponents are generally preferred, especially in final expressions. Positive exponents indicate how many times to multiply the base by itself. Simplifying exponents to positive values contributes to a cleaner and more understandable expression.
In the context of our example, after applying the quotient rule, we managed to convert both original expressions \(a^3/a^{-2}\) and \(b^{-2}/b^{-4}\) to expressions with positive exponents. This conversion is achieved by leveraging the properties of exponents, particularly the rule that allows us to subtract negative exponents.

• \( a^{3 - (-2)} \) becomes \( a^{5} \)
• \( b^{-2 - (-4)} \) becomes \( b^{2} \)

These results clearly illustrate why sticking to positive exponents is beneficial when simplifying mathematical expressions.

Simplifying Expressions

Simplifying expressions is about transforming them into their simplest form, allowing for easier interpretation and manipulation. Here, the simplification process involved using the properties of exponents to unify the base terms and reduce complexity.
After applying the quotient rule and converting to positive exponents, our expression tidied up nicely to \(a^5 b^2\). Simplifying doesn't just avoid zero or negative integers as exponents; it's about achieving an expression that is as straightforward and concise as possible.

• This process ensures that the expression involves only multiplication, avoiding tricky divisions.
• By reducing the expression, you make it easier to read and work with, whether in algebra or practical applications.

Simplifying, therefore, is a crucial step toward a more manageable and mathematically sound expression.