Simplify $(x^{27}y)^{1/3}$: Math Made Easy

Hey there, math enthusiasts! Ever stumbled upon an expression that looks like a jumbled mess of variables and exponents? Well, you're not alone! Today, we're going to dissect one such expression and turn it into something sleek and understandable. Our mission, should we choose to accept it, is to figure out which of the provided options is equivalent to $(x^{27} y)^{\frac{1}{3}}$. Buckle up, because we're about to embark on a mathematical adventure!

The Challenge: Decoding $(x^{27} y)^{\frac{1}{3}}$

Let's start by stating the problem clearly. We need to simplify the expression $(x^{27} y)^{\frac{1}{3}}$. When we look at this, the key thing to notice is the fractional exponent, $\frac{1}{3}$. Remember, fractional exponents are just another way of writing radicals or roots. Specifically, an exponent of $\frac{1}{3}$ means we're taking the cube root. So, we can rewrite our expression as $\sqrt[3]{x^{27} y}$. Now, it looks a little less intimidating, right? But we're not done yet; we want to simplify it as much as possible.

Breaking it Down: The Power of Exponent Rules

The next step involves applying the rules of exponents, which are like the secret sauce for simplifying these kinds of problems. One of the most important rules here is that $(ab)^n = a^n b^n$. This means that if we have a product raised to a power, we can raise each factor in the product to that power individually. Applying this rule to our expression, $\sqrt[3]{x^{27} y}$, we can think of it as $(x^{27} y)^{\frac{1}{3}}$ and distribute the $\frac{1}{3}$ exponent to both $x^{27}$ and $y$. This gives us $(x^{27})^{\frac{1}{3}} \, y^{\frac{1}{3}}$.

Now, we have another exponent rule to play with: $(a^m)^n = a^{mn}$. This rule tells us that when we raise a power to another power, we multiply the exponents. So, for $(x^{27})^{\frac{1}{3}}$, we multiply the exponents 27 and $\frac{1}{3}$. What's 27 times $\frac{1}{3}$? It's 9! Therefore, $(x^{27})^{\frac{1}{3}}$ simplifies to $x^9$.

Taming the Remaining Term: Dealing with $y^{\frac{1}{3}}$

What about the $y^{\frac{1}{3}}$ term? Well, we can't simplify it further as a power of $y$ because there's no other exponent to combine it with. However, remember that an exponent of $\frac{1}{3}$ means taking the cube root. So, $y^{\frac{1}{3}}$ is the same as $\sqrt[3]{y}$. Putting it all together, we have $x^9 \sqrt[3]{y}$.

The Verdict: Choosing the Correct Option

So, after our journey through the land of exponents and roots, we've arrived at the simplified expression: $x^9 \sqrt[3]{y}$. Now, let's look at the options provided:

• A. $x^3(\sqrt[3]{y})$
• B. $x^9(\sqrt[3]{y})$
• C. $x^{27}(\sqrt[3]{y})$
• D. $x^{24}(\sqrt[3]{y})$

Drumroll, please! The correct answer is B. $x^9(\sqrt[3]{y})$. We did it! We successfully simplified the expression and found the equivalent form. This option perfectly matches the result we obtained through our step-by-step simplification process.

Why the Other Options Don't Fit

To solidify our understanding, let's briefly discuss why the other options are incorrect. This will help us avoid similar pitfalls in the future.

• A. $x^3(\sqrt[3]{y})$: This option is incorrect because it seems like the exponent $\frac{1}{3}$ was only applied to the $x^9$ part inside the parenthesis which has an exponent of 27 and simply divided 27 by 9, instead of multiplying 27 by $\frac{1}{3}$. This would be a common mistake if you overlooked the order of operations.
• C. $x^{27}(\sqrt[3]{y})$: This option makes a glaring error. It appears that the $\frac{1}{3}$ exponent wasn't applied to the $x^{27}$ term at all. It's as if the person solving the problem completely ignored the fractional exponent when dealing with the x term. Remember, the exponent outside the parentheses affects every term inside!
• D. $x^{24}(\sqrt[3]{y})$: This option is closer but still misses the mark. It suggests that someone might have subtracted 3 from the exponent 27, instead of correctly multiplying 27 by $\frac{1}{3}$. This kind of error highlights the importance of remembering the rules for exponents and applying them in the correct order.

Mastering Exponents and Radicals: A Recap

Let's take a moment to recap the key concepts we've covered in this mathematical exploration. Understanding these principles will empower you to tackle similar problems with confidence.

• Fractional Exponents: A fractional exponent like $\frac{1}{n}$ represents the nth root. For example, $a^{\frac{1}{3}}$ is the same as $\sqrt[3]{a}$. Understanding this equivalence is crucial for translating between exponential and radical forms.
• Power of a Product Rule: $(ab)^n = a^n b^n$. This rule allows us to distribute an exponent over a product. It's a fundamental tool for breaking down complex expressions into simpler components.
• Power of a Power Rule: $(a^m)^n = a^{mn}$. This rule tells us that when we raise a power to another power, we multiply the exponents. It's essential for simplifying expressions with nested exponents.
• Order of Operations: Always remember the order of operations (PEMDAS/BODMAS). This ensures that you perform calculations in the correct sequence, leading to accurate results.

By mastering these concepts, you'll be well-equipped to conquer expressions involving exponents and radicals. Keep practicing, and you'll become a mathematical wizard in no time!

Real-World Relevance: Why This Matters

Now, you might be thinking,