Simplify Expressions Rewrite In Form Y^n

by ADMIN 41 views

Hey everyone! Today, we're diving into the fascinating world of exponents and how to simplify expressions involving them. Specifically, we'll tackle the problem of rewriting the expression yβˆ’7yβˆ’13{\frac{y^{-7}}{y^{-13}}} in the form yn{y^n}. If you've ever felt a bit puzzled by negative exponents or the rules of dividing powers, don't worry – we're going to break it down step by step. By the end of this guide, you'll not only know how to solve this particular problem but also have a solid understanding of the underlying principles. So, let's jump right in and make exponents a breeze!

Understanding the Basics of Exponents

Before we dive into the nitty-gritty of simplifying expressions, let's make sure we're all on the same page with the fundamentals of exponents. An exponent, simply put, is a way of expressing repeated multiplication. When you see yn{y^n}, it means you're multiplying y{y} by itself n{n} times. For example, y3{y^3} means yΓ—yΓ—y{y \times y \times y}. Now, let's talk about the special case of negative exponents, as they often cause a bit of confusion. A negative exponent, like in yβˆ’7{y^{-7}}, indicates the reciprocal of the base raised to the positive exponent. In other words, yβˆ’7{y^{-7}} is the same as 1y7{\frac{1}{y^7}}. This concept is crucial for simplifying expressions like the one we're tackling today. We also need to remember the quotient rule for exponents, which states that when dividing powers with the same base, you subtract the exponents. That is, yayb=yaβˆ’b{\frac{y^a}{y^b} = y^{a-b}}. This rule is going to be our best friend as we simplify our expression. To really solidify your understanding, think about why this rule works. When you divide, you're essentially canceling out common factors. If you have y5{y^5} divided by y2{y^2}, you're canceling out two y{y}'s from the numerator, leaving you with y3{y^3}. The same principle applies to negative exponents, just with a little extra twist involving reciprocals. So, with these basics in mind, we're well-equipped to tackle our main problem and simplify that expression like pros!

Step-by-Step Simplification of yβˆ’7yβˆ’13{\frac{y^{-7}}{y^{-13}}}

Okay, guys, let's get our hands dirty and simplify the expression yβˆ’7yβˆ’13{\frac{y^{-7}}{y^{-13}}} step by step. Remember, our goal is to rewrite it in the form yn{y^n}, which means we need to find the value of n{n}. The key to solving this lies in applying the quotient rule for exponents, which, as we discussed earlier, states that yayb=yaβˆ’b{\frac{y^a}{y^b} = y^{a-b}}. In our case, we have a=βˆ’7{a = -7} and b=βˆ’13{b = -13}. So, let's plug these values into our formula:

yβˆ’7yβˆ’13=yβˆ’7βˆ’(βˆ’13){ \frac{y^{-7}}{y^{-13}} = y^{-7 - (-13)} }

Now, let's simplify the exponent. Remember that subtracting a negative number is the same as adding its positive counterpart. So, we have:

βˆ’7βˆ’(βˆ’13)=βˆ’7+13{ -7 - (-13) = -7 + 13 }

This simplifies to:

βˆ’7+13=6{ -7 + 13 = 6 }

Therefore, our expression becomes:

yβˆ’7βˆ’(βˆ’13)=y6{ y^{-7 - (-13)} = y^6 }

And there you have it! We've successfully simplified the expression yβˆ’7yβˆ’13{\frac{y^{-7}}{y^{-13}}} to y6{y^6}. This means that n=6{n = 6}. It's pretty cool how the quotient rule allows us to deal with these exponents, even when they're negative. By carefully applying the rule and paying attention to the signs, we can easily transform a seemingly complex expression into a simple and elegant form. The beauty of math lies in these step-by-step processes that lead us to clear and concise solutions. So, let's keep this momentum going and explore some more exponent rules and applications in the next section!

Common Mistakes and How to Avoid Them

Alright, let's talk about some common pitfalls people often stumble into when simplifying expressions with exponents, especially negative ones. Knowing these common mistakes will help you dodge them and get to the correct answer every time. One frequent error is misinterpreting negative exponents. Remember, yβˆ’n{y^{-n}} is not the same as βˆ’yn{-y^n}. The negative sign in the exponent indicates a reciprocal, so yβˆ’n=1yn{y^{-n} = \frac{1}{y^n}}, while βˆ’yn{-y^n} simply means the negative of yn{y^n}. Confusing these two can lead to major errors. Another mistake is messing up the order of operations, particularly when dealing with expressions inside parentheses or multiple exponents. Always remember the golden rule: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Make sure you're simplifying the exponents before you start multiplying or dividing. This is especially crucial when you have nested exponents or expressions with multiple terms. Then, there's the classic mistake of incorrectly applying the quotient rule. Remember, when you're dividing powers with the same base, you subtract the exponents: yayb=yaβˆ’b{\frac{y^a}{y^b} = y^{a-b}}. But it's easy to get mixed up and add them instead, especially when dealing with negative exponents. So, double-check your signs and make sure you're subtracting the exponents in the correct order. Finally, another trap is forgetting to simplify the final answer. Sometimes, you might correctly apply all the exponent rules but end up with an expression that can be further simplified. Always take a final look at your result and see if there are any more simplifications you can make, such as combining like terms or reducing fractions. By being mindful of these common mistakes and taking the time to double-check your work, you can boost your confidence and accuracy when simplifying expressions with exponents. So, let's move on to some more examples and practice these skills!

More Examples and Practice Problems

Okay, guys, let's put our knowledge to the test with some more examples and practice problems. The best way to truly master simplifying expressions with exponents is through practice, practice, practice! We'll start with a couple of examples to warm up and then dive into some problems you can try on your own. Let's kick things off with this one: Simplify {\frac{x^5}{x^{-2}}\. Using the quotient rule, we subtract the exponents: \(5 - (-2) = 5 + 2 = 7}. So, the simplified expression is x7{x^7}. See? Not too shabby! Now, let's try something a bit more challenging: Simplify {\frac{a^{-4}b^3}{a^2b^{-1}}\. Here, we have multiple variables and both positive and negative exponents. But don't worry, we can handle this! We'll apply the quotient rule separately to the \(a} terms and the b{b} terms. For the a{a} terms, we have aβˆ’4a2=aβˆ’4βˆ’2=aβˆ’6{\frac{a^{-4}}{a^2} = a^{-4 - 2} = a^{-6}}. For the b{b} terms, we have b3bβˆ’1=b3βˆ’(βˆ’1)=b3+1=b4{\frac{b^3}{b^{-1}} = b^{3 - (-1)} = b^{3 + 1} = b^4}. Combining these, we get aβˆ’6b4{a^{-6}b^4}. If we want to eliminate the negative exponent, we can rewrite this as b4a6{\frac{b^4}{a^6}}. Fantastic! Now, it's your turn. Here are a few practice problems to try:

  1. Simplify zβˆ’3zβˆ’8{\frac{z^{-3}}{z^{-8}}}
  2. Simplify c2dβˆ’5cβˆ’1d2{\frac{c^2d^{-5}}{c^{-1}d^2}}
  3. Simplify (y4)βˆ’2{(y^4)^{-2}}

Take your time, apply the exponent rules we've discussed, and don't be afraid to make mistakes – that's how we learn! Work through each problem step by step, and remember to double-check your work. Once you've tackled these, you'll be feeling like a true exponent master. Keep practicing, and you'll be simplifying complex expressions with ease in no time!

Real-World Applications of Exponents

So, you might be wondering,