91Ó°ÊÓ

When simplifying expressions that involve exponents, one of the most powerful tools is the 'Power of a Power Rule'. This rule states that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\). In the original exercise, we had the expression \(\frac{p^{-4} q}{r^{-3}}\right)^{-3}\). Applying the 'Power of a Power Rule', each term within the parentheses is raised to the power of -3: \(\frac{(p^{-4})^{-3} \times (q)^{-3}}{(r^{-3})^{-3}}\). This results in: - \((p^{-4})^{-3}\) - \(q^{-3}\) - \((r^{-3})^{-3}\) It's that multiplication of exponents that transforms these terms.

Negative exponents can sometimes be tricky to understand, but they're very straightforward once you get the hang of them. The rule here is: \(a^{-n} = \frac{1}{a^n}\). This means that a negative exponent represents a reciprocal. If you see \(a^{-3}\), you can think of it as \(\frac{1}{a^3}\). In our exercise, we had: - (p^{-4})^{-3} - q^{-3} - (r^{-3})^{-3} Using the negative exponent rule: - \((p^{-4})^{-3}\) becomes \(p^{12}\), because multiplying two negative numbers makes a positive: \(-4 \times -3 = 12\). - \(q^{-3}\) remains the same \(q^{-3}\). - \((r^{-3})^{-3}\) becomes \(r^9\), because multiplying two negative numbers makes a positive: \(-3 \times -3 = 9\).

Fractional exponents are another important aspect of working with powers. A fractional exponent represents both a power and a root. For instance, \(a^{\frac{m}{n}}\) can be written as \(\root{n}\root a^m\), which means the nth root of a raised to the mth power. In our specific exercise, we didn't have any fractional exponents per se, but understanding this concept is crucial when dealing with more complex expressions. Fractional exponents can simplify expressions involving roots. If you encounter something like \((a^{\frac{3}{2}}\) in future exercises, just remember: - It means you take the square (2) root of a and then cube (3) the result. - Alternatively, you can cube a first and then take the square root. Understanding both methods can be very useful!