Problem 40 - 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. $$\left(\frac{p^{-1}}{q^{-5}}\right)^{-2}$$

Step-by-Step Solution

Answer

\(\frac{p^2}{q^{10}}\)

Step 1: Apply the Power of a Quotient Rule

The power of a quotient rule states that \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\). Start by applying this rule to the expression \(\left( \frac{p^{-1}}{q^{-5}} \right)^{-2}\). This gives us: \(\frac{(p^{-1})^{-2}}{(q^{-5})^{-2}}\).

Step 2: Apply the Power Rule

The power rule states that \((a^m)^n = a^{m \cdot n}\). Apply this to both the numerator and the denominator. For the numerator, \((p^{-1})^{-2} = p^{2}\). For the denominator, \((q^{-5})^{-2} = q^{10}\).

Step 3: Rewrite with Positive Exponents

We now have \(\frac{p^{2}}{q^{10}}\). Both exponents are positive, so the expression is already written with positive exponents.

Key Concepts

Power of a Quotient Rule

To start understanding the Power of a Quotient Rule, let’s break down what this rule is all about. The principle behind this is that when you have a division or a quotient raised to an exponent, each element of the quotient gets raised to that exponent. Mathematically, the rule is represented as: \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\). This means, take each part of the fraction separately and apply the power to both the numerator and the denominator.
When you applied this rule to the expression \(\left( \frac{p^{-1}}{q^{-5}} \right)^{-2}\), you individually took the base of each piece of the fraction, \(p^{-1}\) and \(q^{-5}\), giving us \(\frac{(p^{-1})^{-2}}{(q^{-5})^{-2}}\). This looks a bit complicated, but don’t worry, we’ll simplify it later. Utilize this step whenever you see a fraction raised to a power, and break it down into more manageable parts.

Power Rule

The Power Rule is another vital concept when dealing with exponents. It allows us to manage powers raised to other powers efficiently. The rule is stated as \((a^m)^n = a^{m \cdot n}\). Essentially, when you have a power raised to another power, you multiply the exponents together.
In this particular exercise, after applying the Power of a Quotient Rule, you will then use the Power Rule on both parts of the expression. This means for \( (p^{-1})^{-2} \), you multiply the exponents: \(-1 \times -2 = 2\). Similarly, for \( (q^{-5})^{-2} \), it becomes \(-5 \times -2 = 10\). Therefore, the expression simplifies to \(\frac{p^{2}}{q^{10}}\).

• For multiplying exponents: negative signs become positive when multiplied.
• The result is always another exponent, which guides us in rewriting expressions with positive exponents.

Simplifying Expressions

Once you’ve applied the above exponent rules, your next job is to ensure all exponents in the expression are positive. Simplifying expressions to have positive exponents makes them cleaner and aligns with mathematical conventions. This exercise teaches how simplifying can often involve turning confusing combinations into straightforward expressions.
At this stage, the expression \(\frac{p^{2}}{q^{10}}\) is already simplified with positive exponents, meaning no further steps are necessary. If there ever were a negative exponent left, remember the rule: move the base of the exponent from numerator to denominator (or vice versa) and make the exponent positive.
Using the straight path of breaking down and simplifying step by step ensures clarity and understanding. It can seem complex, but sticking to these rules consistently will streamline the process and strengthen your skills in handling exponents.