Simplifying Exponential Expressions: A Step-by-Step Guide

Let's dive into simplifying the exponential expression: $(y^{-4} z^4) * (\frac{3 x^2 y}{z^{-1}})^{-2}$. Our goal is to break it down step by step, ensuring we only use positive exponents in our final answer. This involves understanding and applying the rules of exponents, such as the power of a product, power of a quotient, and negative exponents.

Understanding the Basics of Exponents

Before we start, let's refresh some fundamental exponent rules:

1. Product of Powers: $a^m * a^n = a^{m+n}$
2. Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
3. Power of a Power: $(a^m)^n = a^{m*n}$
4. Power of a Product: $(ab)^n = a^n * b^n$
5. Power of a Quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
6. Negative Exponent: $a^{-n} = \frac{1}{a^n}$
7. Zero Exponent: $a^0 = 1$ (as long as $a \neq 0$)

These rules will be our toolkit as we tackle the simplification process. Understanding how to apply each rule correctly is crucial for achieving the correct result. For instance, recognizing when to apply the power of a power rule versus the product of powers rule can significantly impact the outcome.

Step-by-Step Simplification

Step 1: Address the Outer Exponent

We begin with the given expression: $(y^{-4} z^4) * (\frac{3 x^2 y}{z^{-1}})^{-2}$.

The first thing we'll tackle is the term with the negative exponent outside the parenthesis: $(\frac{3 x^2 y}{z^{-1}})^{-2}$.

Using the rule $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$, we can rewrite this as $(\frac{z^{-1}}{3 x^2 y})^{2}$.

Now, applying the power of a quotient rule, $(\frac{a}{b})^n = \frac{a^n}{b^n}$, we get $\frac{(z^{-1})^2}{(3 x^2 y)^2}$.

Step 2: Simplify the Numerator and Denominator Separately

In the numerator, we have $(z^{-1})^2$. Using the power of a power rule, $(a^m)^n = a^{m*n}$, this simplifies to $z^{-2}$.

In the denominator, we have $(3 x^2 y)^2$. Applying the power of a product rule, $(ab)^n = a^n * b^n$, we get $3^2 * (x^2)^2 * y^2$, which simplifies to $9 x^4 y^2$.

So, our expression now looks like this: $(y^{-4} z^4) * \frac{z^{-2}}{9 x^4 y^2}$.

Step 3: Combine the Terms

Now, let's rewrite the entire expression as a single fraction: $\frac{y^{-4} z^4 * z^{-2}}{9 x^4 y^2}$.

Combining the z terms in the numerator, we have $z^4 * z^{-2}$. Using the product of powers rule, $a^m * a^n = a^{m+n}$, this simplifies to $z^{4 + (-2)} = z^2$.

So, our expression becomes $\frac{y^{-4} z^2}{9 x^4 y^2}$.

Step 4: Eliminate Negative Exponents

We have a negative exponent in the term $y^{-4}$. To eliminate it, we use the rule $a^{-n} = \frac{1}{a^n}$, so $y^{-4} = \frac{1}{y^4}$.

Our expression now looks like $\frac{z^2}{9 x^4 y^2 y^4}$.

Step 5: Simplify Further

Combining the y terms in the denominator, we have $y^2 * y^4$. Using the product of powers rule, $a^m * a^n = a^{m+n}$, this simplifies to $y^{2 + 4} = y^6$.

Thus, our final simplified expression is $\frac{z^2}{9 x^4 y^6}$.

Detailed Breakdown of Each Rule Applied

To ensure clarity, let's revisit each step with a more granular explanation:

• Initial Expression: $(y^{-4} z^4) * (\frac{3 x^2 y}{z^{-1}})^{-2}$
• Applying Negative Exponent Rule to the Fraction: We used $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$ to transform $(\frac{3 x^2 y}{z^{-1}})^{-2}$ into $(\frac{z^{-1}}{3 x^2 y})^{2}$. This step is crucial because it allows us to work with a positive outer exponent.
• Applying Power of a Quotient Rule: We then applied $(\frac{a}{b})^n = \frac{a^n}{b^n}$ to get $\frac{(z^{-1})^2}{(3 x^2 y)^2}$. This separates the numerator and denominator, making it easier to simplify each part individually.
• Simplifying the Numerator with Power of a Power Rule: $(z^{-1})^2$ became $z^{-2}$ using $(a^m)^n = a^{m*n}$. This rule helps simplify exponents that are raised to another power.
• Simplifying the Denominator with Power of a Product Rule: $(3 x^2 y)^2$ became $3^2 * (x^2)^2 * y^2$, which simplifies to $9 x^4 y^2$ using $(ab)^n = a^n * b^n$ and $(a^m)^n = a^{m*n}$. This step expands the denominator, preparing it for further simplification.
• Combining Terms: We rewrote the expression as a single fraction: $\frac{y^{-4} z^4 * z^{-2}}{9 x^4 y^2}$. This prepares us to combine like terms.
• Combining z Terms in the Numerator with Product of Powers Rule: $z^4 * z^{-2}$ became $z^{2}$ using $a^m * a^n = a^{m+n}$. This simplifies the numerator by combining the z terms.
• Eliminating Negative Exponents: $y^{-4}$ became $\frac{1}{y^4}$ using $a^{-n} = \frac{1}{a^n}$. This ensures that all exponents are positive.
• Combining y Terms in the Denominator with Product of Powers Rule: $y^2 * y^4$ became $y^6$ using $a^m * a^n = a^{m+n}$. This simplifies the denominator by combining the y terms.
• Final Simplified Expression: $\frac{z^2}{9 x^4 y^6}$

Common Mistakes to Avoid

When simplifying exponential expressions, it's easy to make mistakes. Here are some common pitfalls to watch out for:

• Incorrectly Applying the Power of a Power Rule: Ensure you multiply the exponents correctly. For example, $(x^2)^3 = x^{2*3} = x^6$, not $x^5$.
• Forgetting to Distribute the Exponent: When raising a product or quotient to a power, remember to apply the exponent to every factor. For example, $(2xy)^2 = 2^2 * x^2 * y^2 = 4x^2y^2$.
• Misunderstanding Negative Exponents: A negative exponent means taking the reciprocal of the base. So, $x^{-2} = \frac{1}{x^2}$, not $-x^2$.
• Adding Exponents When You Should Be Multiplying: Remember, you add exponents when multiplying like bases (e.g., $x^2 * x^3 = x^5$) and multiply exponents when raising a power to a power (e.g., $(x^2)^3 = x^6$).
• Not Simplifying Completely: Always ensure that your final answer has no negative exponents and that all like terms have been combined.

By being mindful of these common errors and practicing regularly, you can improve your accuracy and confidence in simplifying exponential expressions.

Conclusion

Simplifying exponential expressions involves a systematic application of exponent rules. By breaking down the expression into smaller, manageable steps and understanding the rules, you can confidently simplify even complex expressions. Remember to eliminate negative exponents and combine like terms to reach the final simplified form: $\frac{z^2}{9 x^4 y^6}$.

For further learning and practice, check out resources on Khan Academy. They offer excellent explanations and practice problems to help you master exponential expressions.