Simplify $\left[(-80x^2y)(5x^{-6}y^2)\right]^0$: A Quick Guide

Simplifying mathematical expressions can sometimes seem daunting, especially when they involve exponents and variables. However, there are fundamental rules that, once understood, make the process straightforward. In this guide, we will walk through the simplification of the expression $\left[(-80x^2y)(5x^{-6}y^2)\right]^0$. This problem primarily hinges on understanding the zero exponent rule and how to manipulate expressions with exponents. By the end of this article, you'll not only know how to solve this particular problem but also grasp the underlying principles that apply to a wide range of similar expressions.

Understanding the Zero Exponent Rule

The zero exponent rule is a cornerstone in simplifying expressions. This rule states that any non-zero number raised to the power of zero is equal to 1. Mathematically, it is expressed as:

$a^0 = 1$, where $a \neq 0$

This rule applies regardless of how complex the base a is. Whether it's a simple number, a variable, or a complicated expression, as long as the entire base is raised to the power of zero, the result is 1. This is a crucial concept because it dramatically simplifies expressions like the one we are tackling today. Understanding this rule not only helps in simplifying expressions but also provides a foundation for more advanced mathematical concepts. Let's delve deeper into why this rule holds true and how it is derived from other exponent rules. Consider the exponent rule for division:

$\frac{a^m}{a^n} = a^{m-n}$

Now, let's say $m = n$. Then we have:

$\frac{a^m}{a^m} = a^{m-m} = a^0$

Since any number (except zero) divided by itself is 1, we get:

$1 = a^0$

This derivation illustrates why any non-zero number raised to the power of zero equals 1. The zero exponent rule is consistent with the other exponent rules and maintains the logical structure of mathematics. Recognizing this consistency makes it easier to remember and apply the rule correctly. Moreover, understanding the derivation helps to solidify the concept and prevents confusion when encountering more complex problems involving exponents. In essence, the zero exponent rule is not just a shortcut but a fundamental principle rooted in the basic laws of exponents.

Step-by-Step Simplification

Let's simplify the given expression step by step:

$\left[(-80x^2y)(5x^{-6}y^2)\right]^0$

Notice that the entire expression inside the brackets is raised to the power of zero. According to the zero exponent rule, any non-zero expression raised to the power of zero is equal to 1. Therefore,

$\left[(-80x^2y)(5x^{-6}y^2)\right]^0 = 1$

That's it! The expression simplifies to 1. There is no need to evaluate the expression inside the brackets because any non-zero term raised to the power of zero is simply 1. This illustrates the power and simplicity of the zero exponent rule. However, to ensure complete understanding, let's briefly explore what would happen if we were to simplify the expression inside the brackets first. This will reinforce why applying the zero exponent rule directly is the most efficient approach.

First, multiply the constants:

$-80 \times 5 = -400$

Next, multiply the terms with the variable x:

$x^2 \times x^{-6} = x^{2 + (-6)} = x^{-4}$

Then, multiply the terms with the variable y:

$y \times y^2 = y^{1 + 2} = y^3$

So the expression inside the brackets simplifies to:

$-400x^{-4}y^3$

Now, raise this simplified expression to the power of zero:

$(-400x^{-4}y^3)^0 = 1$

As you can see, regardless of whether you simplify the expression inside the brackets first or apply the zero exponent rule directly, the result is the same: 1. This reaffirms the validity and utility of the zero exponent rule. Applying the rule directly saves time and reduces the chances of making errors in the intermediate steps.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's easy to make mistakes. Here are a few common pitfalls to avoid:

1. Forgetting the Zero Exponent Rule: One of the most common mistakes is overlooking the zero exponent rule altogether. Always check if the entire expression (or a part of it) is raised to the power of zero.
2. Misapplying the Rule to Individual Terms: Ensure that the entire expression within the parentheses is raised to the power of zero, not just individual terms. For example, in the expression $(a + b)^0$, the entire term $(a + b)$ is raised to the power of zero, so the result is 1, not $a^0 + b^0$.
3. Incorrectly Simplifying Inside Parentheses: Sometimes, errors occur when simplifying the expression inside the parentheses before applying the zero exponent rule. Double-check each step to ensure accuracy.
4. Ignoring Negative Signs: Pay close attention to negative signs, especially when dealing with exponents. For instance, $(-a)^0$ is 1 if $a \neq 0$, but $-a^0$ is -1.
5. Confusing with Other Exponent Rules: Make sure to differentiate between the zero exponent rule and other exponent rules, such as the product rule, quotient rule, and power rule. Each rule has specific conditions under which it applies.

Avoiding these common mistakes can significantly improve your accuracy and efficiency when simplifying expressions with exponents. Always take a moment to double-check your work and ensure that you have correctly applied the relevant rules.

Practice Problems

To reinforce your understanding, here are a few practice problems:

1. Simplify $(15a^3b^{-2}c^4)^0$
2. Simplify $\left[\frac{2x^5y}{8x^{-3}y^4}\right]^0$
3. Simplify $(-3p^2q^{-1}r)^0$

Solutions:

1. $(15a^3b^{-2}c^4)^0 = 1$
2. $\left[\frac{2x^5y}{8x^{-3}y^4}\right]^0 = 1$
3. $(-3p^2q^{-1}r)^0 = 1$

These practice problems highlight the simplicity of the zero exponent rule. No matter how complex the expression inside the parentheses, if the entire expression is raised to the power of zero, the result is always 1.

Conclusion

In summary, simplifying $\left[(-80x^2y)(5x^{-6}y^2)\right]^0$ is a straightforward application of the zero exponent rule. The expression simplifies directly to 1, highlighting the importance and simplicity of this rule. By understanding and applying the zero exponent rule correctly, you can efficiently simplify a wide variety of mathematical expressions. Remember to avoid common mistakes and always double-check your work to ensure accuracy. With practice, simplifying expressions with exponents will become second nature.

For further learning on exponent rules, you might find this resource helpful: Exponent Rules - Math is Fun