Problem 39 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\left(\frac{r^{-2}}{s^{-5}}\right)^{-3}$$

Step-by-Step Solution

Answer

\( \frac{r^6}{s^{15}} \)

Step 1: Apply the Power of a Quotient Rule

We start with \( \left(\frac{r^{-2}}{s^{-5}}\right)^{-3} \). The rule for a power of a quotient, \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \), allows us to distribute the outer exponent to both the numerator and the denominator. Therefore, we have \( (r^{-2})^{-3} / (s^{-5})^{-3} \).

Step 2: Simplify the Exponents

Now, we use the rule \((a^m)^n = a^{m \cdot n}\) for the exponents to simplify each term. For the numerator: \((r^{-2})^{-3} = r^{(-2)\cdot(-3)} = r^6\). For the denominator: \((s^{-5})^{-3} = s^{(-5)\cdot(-3)} = s^{15}\).

Step 3: Write the Expression with Positive Exponents

After simplification, the expression \(\frac{r^6}{s^{15}}\) already has only positive exponents, as required.

Key Concepts

Power of a Quotient Rule

The Power of a Quotient Rule is a handy tool when dealing with exponents in fractions. This rule states that \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). Essentially, when you have a power applied to a fraction, you can distribute this power to both the numerator and the denominator separately.
Imagine you have \( \left(\frac{r^{-2}}{s^{-5}}\right)^{-3} \). Applying the Power of a Quotient Rule means taking the \( -3 \) exponent and distributing it to both \( r^{-2} \) and \( s^{-5} \). So it becomes \( (r^{-2})^{-3} / (s^{-5})^{-3} \).
This step is crucial for simplifying complex expressions as it breaks down the problem into more manageable parts. Remember, this is only applicable when a single external exponent is multiplying a quotient, helping you treat each part individually.

Simplifying Exponents

Simplifying exponents involves using certain rules to make expressions neater and easier to work with. In our example, once the exponent \( -3 \) is distributed to both the terms in the fraction, each of these terms needs to be simplified.
The rule we'll use is \( (a^m)^n = a^{m \cdot n} \). This means you multiply the exponents to simplify. So for the term \( (r^{-2})^{-3} \), it becomes \( r^{(-2) \times (-3)} = r^6 \). Similarly, for \( (s^{-5})^{-3} \), it turns into \( s^{(-5) \times (-3)} = s^{15} \).
The important thing to remember is that multiplying two negative exponents results in a positive exponent. This is a key step in transforming your expression toward having all positive exponents, which makes it simpler and often required in solutions.

Positive Exponents

Positive exponents are simpler and more straightforward to work with than negative exponents. This is why final expressions in mathematics are often required to have only positive exponents.
If you start with a negative exponent, as in \( r^{-2} \) or \( s^{-5} \), and after applying power rules you end up with a positive exponent, your task is complete. Remember, a negative exponent indicates the reciprocal of the base raised to the opposite positive power, but simplifying changes these negatives to positives.
For example, \( r^{-2} \) becomes \( r^6 \) and \( s^{-5} \) becomes \( s^{15} \) after simplification. So the expression ultimately becomes \( \frac{r^6}{s^{15}} \), which successfully contains only positive exponents. Transitioning to positive exponents not only satisfies problem requirements but also paves way for easier calculations in further steps.