Understanding exponent rules is crucial for simplifying expressions effectively. When faced with a complex exponential expression, it's essential to apply multiple exponent rules in combination to achieve full simplification. This process can be approached as a checklist rather than a strict step-by-step method, allowing for flexibility in how you tackle the problem.
Consider the expression \(3x^{-5}\) squared and \(-2x^{4}\) cubed. The first step is to check for powers raised to other powers. In this case, you can apply the power rule, which states that when raising a power to another power, you multiply the exponents. Thus, \( (3x^{-5})^2 \) becomes \( 3^2 \cdot x^{-5 \cdot 2} = 9x^{-10} \) and \( (-2x^{4})^3 \) becomes \( (-2)^3 \cdot x^{4 \cdot 3} = -8x^{12} \). After this, ensure that all numbers with exponents are evaluated, leading to \( 9x^{-10} \) and \(-8x^{12}\).
Next, check for like bases that are multiplied or divided. Here, you have \( x^{-10} \) and \( x^{12} \). Using the product rule, which allows you to add exponents when multiplying like bases, you combine these to get \( x^{-10 + 12} = x^{2} \). Now, multiply the coefficients: \( 9 \times -8 = -72 \). The final simplified expression is \( -72x^{2} \).
In another example, consider the expression \( \frac{x^{2}y^{7}}{x^{5}y^{4}} \) raised to the power of \(-1\). Instead of distributing the negative exponent immediately, simplify the fraction first. Using the quotient rule, \( \frac{x^{2}}{x^{5}} \) simplifies to \( x^{2-5} = x^{-3} \) and \( \frac{y^{7}}{y^{4}} \) simplifies to \( y^{7-4} = y^{3} \). This gives you \( \frac{y^{3}}{x^{3}} \) raised to the power of \(-1\).
Now, apply the negative exponent rule, which states that a negative exponent indicates a reciprocal. Thus, \( \frac{y^{3}}{x^{3}} \) becomes \( \frac{x^{3}}{y^{3}} \). Ensure that all operations are performed, and since there are no further simplifications needed, the expression is fully simplified.
In summary, when simplifying exponential expressions, systematically apply the exponent rules, evaluate numbers, combine like bases, and address any negative exponents to achieve the simplest form. This approach not only clarifies the process but also reinforces the understanding of how exponent rules interact in various scenarios.
