Simplify Exponent Expressions: A Step-by-Step Guide

Hey guys! Today, we're going to break down a common algebra problem that involves exponents and fractional powers. We'll be looking at the expression ${\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}}$ and figuring out which of the given options is equivalent to it. Exponents might seem intimidating at first, but don't worry, we'll take it step by step and make sure you understand the logic behind each operation. Our goal is not just to find the correct answer, but also to equip you with the tools and knowledge to tackle similar problems in the future. So, let's dive in and unlock the secrets of exponents together!

Initial Expression: ${\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}}$

Let's start by revisiting the expression we're working with: ${\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}}$. This might look a bit complex at first glance, but it's actually quite manageable if we break it down into smaller, more digestible parts. The key here is to remember the fundamental rules of exponents. When you see something like ${x^{\frac{4}{3}}}$, think of it as 'x' raised to the power of 4/3. The fraction in the exponent indicates both a power and a root. The numerator (4 in this case) is the power, and the denominator (3 in this case) is the root. So, ${x^{\frac{4}{3}}}$ can also be thought of as the cube root of x raised to the fourth power. We'll be using these concepts as we simplify the expression. The expression also includes another term, ${x^{\frac{2}{3}}}$ and the entire quantity inside the parenthesis is raised to the power of ${\frac{1}{3}}$. Let's look at how to deal with the terms inside the parenthesis first.

Step 1: Simplifying Inside the Parentheses

The first step to simplifying this expression involves focusing on what's inside the parentheses: ${x^{\frac{4}{3}} x^{\frac{2}{3}}}$. Here, we have two terms with the same base ('x') being multiplied together. Remember the rule for multiplying exponents with the same base: you add the exponents. In other words, ${x^a * x^b = x^{a+b}}$. Applying this rule to our expression, we get ${x^{\frac{4}{3}} x^{\frac{2}{3}} = x^{\frac{4}{3} + \frac{2}{3}}}$. Now, we just need to add the fractions in the exponent. Since they have the same denominator, it's pretty straightforward: ${\frac{4}{3} + \frac{2}{3} = \frac{4+2}{3} = \frac{6}{3}}$. So, our expression inside the parentheses simplifies to ${x^{\frac{6}{3}}}$. But we're not done yet! We can further simplify the fraction ${\frac{6}{3}}$. Dividing 6 by 3 gives us 2. Therefore, ${x^{\frac{6}{3}}}$ is the same as ${x^2}$. Now, we've simplified the expression inside the parentheses to a much cleaner form. This makes the next step of dealing with the outer exponent a lot easier. Remember, the key here was recognizing the rule for multiplying exponents with the same base and then simplifying the resulting fraction.

Step 2: Applying the Outer Exponent

Okay, so we've simplified the inside of the parentheses to ${x^2}$. Now, let's bring back the outer exponent: ${\left(x^2\right)^{\frac{1}{3}}}$. This is where another important rule of exponents comes into play: 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 represented as ${(x^a)^b = x^{a*b}}$. Applying this rule to our expression, we multiply the exponents 2 and ${\frac{1}{3}}$. So, we have ${x^{2 * \frac{1}{3}}}$. Multiplying 2 by ${\frac{1}{3}}$ is the same as ${\frac{2}{1} * \frac{1}{3}}$, which equals ${\frac{2}{3}}$. Therefore, ${\left(x^2\right)^{\frac{1}{3}}}$ simplifies to ${x^{\frac{2}{3}}}$. And there you have it! We've successfully simplified the entire expression by applying the rules of exponents step by step. We first simplified the expression inside the parentheses by adding the exponents and then applied the power of a power rule to handle the outer exponent. This methodical approach is crucial for tackling more complex problems involving exponents.

Step 3: Identifying the Equivalent Expression

Great job, guys! We've successfully simplified the original expression ${\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}}$ to ${x^{\frac{2}{3}}}$. Now, the final step is to match our simplified expression with the given options to identify the equivalent one. Remember, the options were:

A. ${x^{\frac{2}{9}}}$ B. ${x^{\frac{2}{3}}}$ C. ${x^{\frac{8}{27}}}$ D. ${x^{\frac{7}{3}}}$

By comparing our simplified expression, (x^{\frac{2}{3}}, with the options, it's clear that option B, (x^{\frac{2}{3}}, is the correct answer. We've gone from a seemingly complex expression to a straightforward solution by applying the fundamental rules of exponents. This process highlights the importance of understanding these rules and how they can be used to simplify and solve algebraic problems. You can see how breaking down the problem into manageable steps made it much easier to solve. Always remember to look for opportunities to simplify within parentheses first and then apply the power of a power rule when necessary.

Common Mistakes to Avoid

Before we wrap up, let's quickly touch on some common mistakes people make when dealing with expressions like this. Recognizing these pitfalls can help you avoid them in your own problem-solving.

• Mistake 1: Forgetting the order of operations. Just like with any mathematical expression, the order of operations (PEMDAS/BODMAS) is crucial. Make sure you simplify inside parentheses first before applying outer exponents. In our example, simplifying ${x^{\frac{4}{3}} x^{\frac{2}{3}}}$ before dealing with the ${\frac{1}{3}}$ exponent was key.
• Mistake 2: Incorrectly adding exponents when multiplying terms with the same base. Remember, when multiplying terms with the same base, you add the exponents, not multiply them. So, ${x^{\frac{4}{3}} x^{\frac{2}{3}}}$ becomes {x^{\frac{4}{3} + \frac{2}{3}}\, not \(x^{\frac{4}{3} * \frac{2}{3}}}.
• Mistake 3: Misapplying the power of a power rule. The power of a power rule states that ${(x^a)^b = x^{a*b}}$. A common mistake is to add the exponents instead of multiplying them. Make sure you're clear on this distinction.
• Mistake 4: Not simplifying fractions in the exponents. Always simplify fractions whenever possible. In our case, ${\frac{6}{3}}$ simplifies to 2, which makes the expression much easier to work with.
• Mistake 5: Getting confused by fractional exponents. Remember that a fractional exponent represents both a power and a root. For example, ${x^{\frac{a}{b}}}$ is the same as the b-th root of x raised to the power of a. Keeping this in mind can help you better understand and manipulate these expressions.

By being aware of these common mistakes, you can approach similar problems with greater confidence and accuracy.

Conclusion: Mastering Exponents

Alright, guys, we've reached the end of our journey through this exponent problem! We successfully simplified the expression ${\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}}$ and found that it's equivalent to ${x^{\frac{2}{3}}}$. More importantly, we've reinforced some key concepts and rules about exponents that will help you in a wide range of mathematical problems. Remember, the key to mastering exponents is understanding the rules and applying them systematically. Don't be afraid to break down complex expressions into smaller, more manageable steps. Practice is also essential. The more you work with exponents, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep those mathematical muscles flexing! You've got this! If you have any questions or want to explore more exponent problems, feel free to ask. Happy problem-solving!