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Negative exponents can be confusing, but they follow a simple rule. When you see a negative exponent, it means you're dealing with the reciprocal of the base. For example, if you have a base with a negative exponent, like \(a^{-n}\), it can be turned into a positive exponent by moving the base to the denominator: \(a^{-n} = \frac{1}{a^n}\). This changes the negative exponent into a positive one.

In the given example, starting from \(p^{-7} q^{-4}\), each base with a negative exponent means we would take their reciprocals, converting the negative exponents into positive ones. However, we'll apply other rules to simplify further. Understanding negative exponents is crucial because it allows you to rewrite expressions in more manageable forms.

The power of a power rule is straightforward but very useful. This rule states that \((a^m)^n = a^{m \times n}\). Essentially, you multiply the exponents when raising a power to another power. This rule greatly simplifies expressions where you have one exponent raised to another exponent.

For our exercise, we start with \((p^{-7} q^{-4})^{-2}\). Applying the power of a power rule, we multiply the exponents: \(p^{-7 \times -2}\) and \(q^{-4 \times -2}\). This multiplication results in new exponents for each base, making the expression easier to handle.

Multiplying exponents is the next step after applying the power of a power rule. When we multiply exponents, we're simplifying the expression even further. For the given exercise, it involves straightforward multiplication.

Here's how it works:

• Multiply \(-7\) and \(-2\) for the exponent of \(p\): \(-7 \times -2 = 14\). So, \(p^{-7 \times -2} = p^{14}\).
• Do the same for \(q\): \(-4 \times -2 = 8\). Thus, \(q^{-4 \times -2} = q^{8}\).

Now, the original expression \((p^{-7} q^{-4})^{-2}\) simplifies to \(p^{14} q^{8}\). This final expression uses positive exponents only, as required.