Simplifying Rational Exponents A Step-by-Step Guide

Hey guys! Let's dive into the world of rational exponents and tackle a common sticking point: simplifying expressions. Today, we're going to break down a specific problem and make sure you understand exactly why things work the way they do. No more exponent confusion – let's get started!

The Problem at Hand

We're faced with simplifying the following expression:

$$\frac{\left(x^5y^{8/3}\right)^{1/4}}{x^{1/16}y^{7/24}}$$

Our goal is to get this into its simplest form, using the rules of exponents. The initial attempt resulted in $x^{20}y{16/7}$, but it turns out that's not quite right. So, let's dissect what went wrong and how to get to the correct answer.

Understanding the Rules of Exponents

Before we jump into the solution, let's quickly recap the key exponent rules we'll be using. These rules are the bedrock of simplifying expressions with exponents, so make sure you're comfortable with them.

• Power of a Power: When you raise a power to another power, you multiply the exponents: $(x^a)^b = x^{a*b}$
• Product of Powers: When multiplying powers with the same base, you add the exponents: $x^a * x^b = x^{a+b}$
• Quotient of Powers: When dividing powers with the same base, you subtract the exponents: $\frac{x^a}{x^b} = x^{a-b}$
• Negative Exponents: A negative exponent indicates a reciprocal: $x^{-a} = \frac{1}{x^a}$
• Fractional Exponents: A fractional exponent represents a root: $x^{a/b} = \sqrt[b]{x^a}$

These rules might seem a bit abstract on their own, but they become much clearer when we apply them to actual problems. Let's see how they work in our case.

Step-by-Step Solution

Okay, let's break down the simplification process step-by-step. We'll take it nice and slow to make sure everyone's on the same page.

Step 1: Applying the Power of a Power Rule

The first thing we need to do is tackle the numerator. We have $(x^5y^{8/3})^{1/4}$. This means we need to apply the power of a power rule to both the $x^5$ and the $y^{8/3}$ terms.

• For $x^5$, we have $(x^5)^{1/4}$. Multiplying the exponents, we get $x^{5*(1/4)} = x^{5/4}$.
• For $y^{8/3}$, we have $(y^{8/3})^{1/4}$. Multiplying the exponents, we get $y^{(8/3)*(1/4)} = y^{8/12}$. We can simplify the fraction 8/12 to 2/3, so we have $y^{2/3}$.

So, the numerator simplifies to $x^{5/4}y^{2/3}$.

Step 2: Rewriting the Expression

Now, let's rewrite the entire expression with the simplified numerator:

$$\frac{x^{5/4}y^{2/3}}{x^{1/16}y^{7/24}}$$

This looks much more manageable already! We've gotten rid of the parentheses and have individual terms in the numerator and denominator.

Step 3: Applying the Quotient of Powers Rule

Now, we can use the quotient of powers rule. Remember, this means we subtract the exponents of the terms with the same base.

• For the $x$ terms, we have $\frac{x^{5/4}}{x^{1/16}}$. Subtracting the exponents, we get $x^{(5/4)-(1/16)}$. To subtract these fractions, we need a common denominator, which is 16. So, we have $x^{(20/16)-(1/16)} = x^{19/16}$.
• For the $y$ terms, we have $\frac{y^{2/3}}{y^{7/24}}$. Subtracting the exponents, we get $y^{(2/3)-(7/24)}$. Again, we need a common denominator, which is 24. So, we have $y^{(16/24)-(7/24)} = y^{9/24}$. We can simplify the fraction 9/24 to 3/8, so we have $y^{3/8}$.

Step 4: The Simplified Expression

Putting it all together, our simplified expression is:

$$x^{19/16}y^{3/8}$$

And there you have it! We've successfully simplified the expression using the rules of exponents.

Where Did the Initial Attempt Go Wrong?

So, what happened in the initial attempt that led to the incorrect answer of $x^{20}y{16/7}$? It seems like there might have been a mix-up in applying the power of a power rule and the quotient of powers rule.

Specifically, multiplying the exponents inside the parentheses in the numerator seems to be the source of the error. Remember, the power of a power rule states that you multiply exponents when raising a power to another power, not when terms are already multiplied within the base.

Additionally, there may have been an error in the subtraction of the exponents when applying the quotient of powers rule. It's crucial to find a common denominator before subtracting fractions, and it's easy to make a mistake if you rush through this step.

Common Mistakes to Avoid

Simplifying expressions with rational exponents can be tricky, and it's easy to make mistakes. Here are a few common pitfalls to watch out for:

• Forgetting the Power of a Power Rule: This is a big one! Make sure you remember to multiply the exponents when raising a power to another power.
• Incorrectly Applying the Quotient of Powers Rule: Remember to subtract the exponents when dividing terms with the same base. Don't add them!
• Not Finding a Common Denominator: When adding or subtracting fractions (which you'll often need to do with rational exponents), always find a common denominator first.
• Rushing Through the Steps: It's tempting to try and simplify everything at once, but it's much safer to break the problem down into smaller, manageable steps. This will help you avoid errors.

By being mindful of these common mistakes, you can significantly improve your accuracy when simplifying expressions with rational exponents.

Practice Makes Perfect

The best way to master simplifying expressions with rational exponents is to practice, practice, practice! Work through plenty of examples, and don't be afraid to make mistakes. Every mistake is a learning opportunity.

Try working through similar problems, changing the exponents and coefficients to challenge yourself. The more you practice, the more comfortable you'll become with the rules of exponents, and the easier it will be to simplify complex expressions.

Conclusion

Simplifying expressions with rational exponents might seem daunting at first, but by understanding the rules of exponents and breaking the problem down into manageable steps, you can conquer even the trickiest expressions. Remember to apply the power of a power rule, the quotient of powers rule, and always double-check your work.

So, the correct simplification of the expression $\frac{\left(x^5y{8/3}\right)^{1/4}}{x{1/16}y^{7/24}}$ is $x^{19/16}y{3/8}$. Keep practicing, and you'll become a pro at simplifying rational exponents in no time!

If you guys have any questions or want to tackle more examples, just let me know in the comments. Happy simplifying!