Simplifying The Expression: Y⁻⁴(y³)², Step-by-Step

Hey math enthusiasts! Let's dive into simplifying the expression *y⁻⁴(y³)². * This might look a little intimidating at first, but trust me, breaking it down step-by-step makes it super manageable. We're going to use some fundamental rules of exponents to make this problem a breeze. So, grab your pencils, and let's get started! Our goal is to transform this expression into its simplest form, which will give us a better understanding of how exponents work and how to manipulate them. Are you ready to see how simple it is?

Understanding the Basics of Exponents

Before we jump into the problem, let's quickly recap some key exponent rules. Remember, these rules are the building blocks for simplifying expressions like ours. If you're a bit rusty on these, don't worry – a quick refresher will get you back on track. This will help you to understand what we are going to do later.

First up, we have the power of a power rule. This rule states that when you raise a power to another power, you multiply the exponents. Mathematically, this is expressed as (xᵃ)ᵇ = xᵃᵇ. For example, (2²)³ equals 2 raised to the power of (2*3), which is 2⁶. This rule is going to be incredibly useful for our expression, *y⁻⁴(y³)². * We have to apply it properly so that we do not make any mistakes.

Next, we have the product of powers rule. When you multiply two terms with the same base, you add the exponents. This is written as xᵃ * xᵇ = xᵃ⁺ᵇ. For instance, 2² * 2³ equals 2 raised to the power of (2+3), which is 2⁵. Also, we will need the negative exponent rule, which says that x⁻ᵃ = 1/xᵃ. This rule is especially important because it helps us deal with the negative exponent in our expression. And finally, the zero exponent rule: any non-zero number raised to the power of zero is always equal to 1 (x⁰ = 1). Keep these rules in mind as we work through y⁻⁴(y³)². Now that we have our core concepts down, let's begin simplifying the expression.

Step-by-Step Simplification

Alright, guys, let's get to the fun part: simplifying *y⁻⁴(y³)². * We'll break it down into manageable steps to make sure we don't miss anything. Here is what we are going to do. We will use the rules that we have introduced previously.

Step 1: Simplify (y³)² using the Power of a Power Rule. First, we address the term *(y³)². * Using the power of a power rule, we multiply the exponents: 3 * 2 = 6. This transforms (y³) ² into y⁶. This is a crucial first step; if you get this wrong, the rest of the problem will be wrong as well. At the end of this step, our expression becomes y⁻⁴ * y⁶. Now, let's go to step two.

Step 2: Combine y⁻⁴ * y⁶ using the Product of Powers Rule. Now, we have y⁻⁴ * y⁶. According to the product of powers rule, we add the exponents. So, we add -4 and 6, which gives us 2. This means *y⁻⁴ * y⁶ = y². * This is nearly our final answer! We are almost at the end. At the end of this step, we should have y². Now, we've got a simplified form.

Therefore, the simplified form of the expression y⁻⁴(y³) ² is *y². * Easy peasy, right?

Final Answer and Explanation

So, after all that, we've arrived at the simplified form of y⁻⁴(y³) ², which is y². Let's recap what we did to get there, just to make sure everything clicks. We started with y⁻⁴(y³) ². First, we used the power of a power rule to simplify (y³) ² to y⁶. Then, we combined y⁻⁴ * y⁶ using the product of powers rule, which resulted in y². Understanding these steps is important because these principles can be applied to many other problems.

Essentially, the expression simplifies because we're combining powers of the same base. When we multiply powers, we add the exponents. When we raise a power to a power, we multiply the exponents. It's all about keeping track of those exponent rules. You are basically doing the opposite of what is being expressed, in terms of complexity. Now that you have learned about the procedure, you can start doing other similar problems to master the topic.

Common Mistakes to Avoid

When simplifying exponential expressions, there are a few common pitfalls that can trip you up. Knowing these mistakes can help you avoid them, making your problem-solving process smoother. This is the last part. Here, we can explore some common mistakes.

One common mistake is incorrectly applying the power of a power rule. Remember, it's about multiplying the exponents, not adding them. For example, some people might mistakenly think that (y³) ² equals y⁵ instead of y⁶. Always double-check that you're multiplying the exponents correctly.

Another frequent mistake is mishandling negative exponents. A negative exponent indicates that the term should be moved to the denominator (or the denominator to the numerator), as in the expression. If you're not careful, you might end up with a negative exponent in your final answer when it should be positive, or vice versa. Keep in mind the negative exponent rule x⁻ᵃ = 1/xᵃ. So, be careful when applying this rule.

Finally, don't forget to apply the rules of exponents in the correct order. Often, you'll need to use multiple rules in a single problem, and the order in which you apply them can make a difference. Always start by addressing any parentheses or nested exponents, then move on to multiplication and division, and finally, addition and subtraction. By being aware of these common mistakes, you'll be well on your way to simplifying exponential expressions with confidence and accuracy. Keep practicing and you will get better!