Equivalent Expressions For Cube Root Of X To The Power Of 5 Times Y

Hey guys! Today, we're diving into the fascinating world of algebra and exponents to figure out which expression is equivalent to $\sqrt[3]{x^5 y}$. This type of problem is super common in mathematics, especially in algebra and calculus, so getting a good handle on it is really important. Let's break it down step by step so you can ace these types of questions every time!

Decoding the Expression $\sqrt[3]{x^5 y}$

Okay, so we have the expression $\sqrt[3]{x^5 y}$. The first thing we need to recognize is that this involves a cube root (the little '3' in the radical) and some variables raised to powers. Remember, guys, a root is just the inverse operation of an exponent. Specifically, a cube root is the inverse of cubing something (raising it to the power of 3). To really nail this, let's refresh some key concepts about radicals and exponents.

Radicals and Exponents: A Quick Refresher

Think of a radical like a house that protects a value. In our case, the house is the cube root symbol, $\sqrt[3]{ }$, and what's inside is $x^5 y$. The index of the radical (the '3' in our case) tells us what root we're taking. So, $\sqrt[3]{ }$ means we're looking for a number that, when multiplied by itself three times, gives us the expression inside the radical.

Exponents, on the other hand, tell us how many times to multiply a number by itself. For example, $x^5$ means $x$ multiplied by itself five times: $x * x * x * x * x$. Now, here's the cool part: radicals and exponents are closely related! We can rewrite radicals as fractional exponents, which is super helpful for simplifying expressions.

The Magic Trick: Converting Radicals to Fractional Exponents

This is where the magic happens, guys! The general rule is: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. In plain English, the nth root of $a$ raised to the power of $m$ is the same as $a$ raised to the power of $m$ divided by $n$. Let's see how this applies to our problem.

In our expression, $\sqrt[3]{x^5 y}$, we have a cube root (so $n = 3$). Inside the radical, we have $x^5$ and $y$. Notice that $y$ can be thought of as $y^1$. Now we can rewrite the expression using fractional exponents. For $x^5$ inside the cube root, we get $x^{\frac{5}{3}}$. For $y$ (or $y^1$) inside the cube root, we get $y^{\frac{1}{3}}$.

Putting It All Together: Rewriting $\sqrt[3]{x^5 y}$

Now that we've converted the radical into fractional exponents, we can rewrite the entire expression. Remember, the cube root applies to both $x^5$ and $y$. So, we have:

$\sqrt[3]{x^5 y} = (x^5 y)^{\frac{1}{3}}$

Using the rule that $(ab)^n = a^n b^n$, we can distribute the exponent $\frac{1}{3}$ to both $x^5$ and $y$:

$(x^5 y)^{\frac{1}{3}} = x^{5 * \frac{1}{3}} * y^{\frac{1}{3}} = x^{\frac{5}{3}} y^{\frac{1}{3}}$

So, there you have it! The expression $\sqrt[3]{x^5 y}$ is equivalent to $x^{\frac{5}{3}} y^{\frac{1}{3}}$.

Analyzing the Options

Now, let's look at the options we were given and see which one matches our simplified expression:

• $x^{\frac{5}{3}} y$
• $x^{\frac{5}{3}} y^{\frac{1}{3}}$
• $x^{\frac{3}{5}} y$
• $x^{\frac{3}{5}} y^3$

By comparing these options with our simplified expression, $x^{\frac{5}{3}} y^{\frac{1}{3}}$, it's clear that the correct answer is the second option: $x^{\frac{5}{3}} y^{\frac{1}{3}}$.

Let's quickly go through why the other options are incorrect:

• $x^{\frac{5}{3}} y$: This is close, but it's missing the fractional exponent on the $y$. It only takes the cube root of $x^5$ and not $y$.
• $x^{\frac{3}{5}} y$: This one has the exponent on $x$ flipped ($\frac{3}{5}$ instead of $\frac{5}{3}$) and is also missing the fractional exponent on $y$.
• $x^{\frac{3}{5}} y^3$: This option has both exponents incorrect. The exponent on $x$ is flipped, and the exponent on $y$ is 3 instead of $\frac{1}{3}$.

Key Takeaways and Tips for Success

Alright, guys, let's recap the key things we learned and some tips for tackling similar problems:

1. Master the Conversion: The ability to convert between radicals and fractional exponents is crucial. Remember the rule: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Practice this until it becomes second nature!
2. Distribute Exponents Correctly: When you have an expression inside parentheses raised to a power, make sure to distribute the exponent to every term inside the parentheses. For example, $(xy)^n = x^n y^n$.
3. Simplify Step by Step: Break down the problem into smaller, manageable steps. Don't try to do everything at once. Simplify the exponents and radicals one at a time.
4. Double-Check Your Work: Always go back and check your answer, especially in timed tests. Make sure you haven't made any silly mistakes with the exponents or fractions.
5. Practice Makes Perfect: The more problems you solve, the better you'll become at recognizing patterns and applying the rules. So, keep practicing!

Real-World Applications and Why This Matters

You might be wondering,