Decoding Equivalent Expressions Which Expression Is Equivalent To $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$

Hey math enthusiasts! Ever stumbled upon a seemingly complex math problem that just makes you scratch your head? Well, we've got one of those today. Let's dive into this fascinating mathematical puzzle and break it down step by step. Our mission, should we choose to accept it, is to figure out which expression is equivalent to $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$. Sounds like fun, right? Let’s explore the options together and unravel this mathematical conundrum!

Understanding the Problem

Before we jump into solving this, let’s make sure we understand the key concepts. The expression we're dealing with involves roots and fractions, which might seem intimidating at first, but don't worry, we'll tackle it together. We need to simplify $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$ and then compare it to the given options. Remember, the goal here is to find an equivalent expression, meaning one that has the same value. Think of it like finding different ways to say the same thing – in math!

Roots and Radicals

Let's quickly refresh what roots are. A root, also known as a radical, is the inverse operation of raising a number to a power. For example, the square root of 9 ($\sqrt{9}$) is 3 because 3 squared ($3^2$) is 9. Similarly, the cube root of 8 ($\sqrt[3]{8}$) is 2 because 2 cubed ($2^3$) is 8. In our problem, we have a fourth root ($\sqrt[4]{6}$) and a cube root ($\sqrt[3]{2}$). These tell us what number, when raised to the fourth or third power, respectively, will give us the number inside the root.

Fractions and Simplification

We also need to remember how fractions work. A fraction represents a part of a whole, and in our case, it's a division. To simplify fractions involving roots, we often look for ways to combine the roots or rewrite them in a more manageable form. This might involve finding common denominators for the fractional exponents or rationalizing the denominator (getting rid of the root in the denominator). We'll see how these techniques come into play as we solve the problem.

Why This Matters

Understanding how to manipulate and simplify expressions with roots and fractions isn't just an academic exercise. These skills are crucial in various fields, including engineering, physics, and computer science. They allow us to solve complex problems, model real-world phenomena, and make accurate predictions. Plus, it's super satisfying when you can take a complicated-looking expression and simplify it into something elegant and easy to understand!

Breaking Down the Expression $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$

Okay, guys, let's get our hands dirty and dive into simplifying $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$. The key here is to rewrite the roots as fractional exponents. Remember, a root can be expressed as a fractional power. For example, $\sqrt[n]{a}$ is the same as $a^{\frac{1}{n}}$. This transformation will allow us to use the rules of exponents to simplify the expression.

Converting Roots to Fractional Exponents

So, let's rewrite our expression using fractional exponents. $\sqrt[4]{6}$ becomes $6^{\frac{1}{4}}$, and $\sqrt[3]{2}$ becomes $2^{\frac{1}{3}}$. Now our expression looks like this: $\frac{6^{\frac{1}{4}}}{2^{\frac{1}{3}}}$. See? We've already made progress! It might still look a bit intimidating, but we're on the right track.

Finding a Common Denominator

To further simplify, we want to have a common denominator for the fractional exponents. This will allow us to combine the terms more easily. The denominators we have are 4 and 3, so their least common multiple is 12. This means we want to rewrite the exponents with a denominator of 12.

To do this, we multiply the numerator and denominator of each fraction by the appropriate number to get a denominator of 12. For $6^{\frac{1}{4}}$, we multiply the exponent by $\frac{3}{3}$, giving us $6^{\frac{3}{12}}$. For $2^{\frac{1}{3}}$, we multiply the exponent by $\frac{4}{4}$, giving us $2^{\frac{4}{12}}$. Now our expression looks like this: $\frac{6^{\frac{3}{12}}}{2^{\frac{4}{12}}}$.

Combining the Terms

Now that we have a common denominator, we can rewrite the expression using the 12th root. Remember, $a^{\frac{m}{n}}$ is the same as $\sqrt[n]{a^m}$. So, $6^{\frac{3}{12}}$ becomes $\sqrt[12]{6^3}$, and $2^{\frac{4}{12}}$ becomes $\sqrt[12]{2^4}$. Our expression now looks like $\frac{\sqrt[12]{6^3}}{\sqrt[12]{2^4}}$.

We can simplify this further by calculating $6^3$ and $2^4$. $6^3 = 6 \times 6 \times 6 = 216$, and $2^4 = 2 \times 2 \times 2 \times 2 = 16$. So our expression is now $\frac{\sqrt[12]{216}}{\sqrt[12]{16}}$. Since both terms are under the same root, we can combine them into a single 12th root: $\sqrt[12]{\frac{216}{16}}$.

Simplifying the Fraction

Let's simplify the fraction inside the root. $\frac{216}{16}$ can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 8. So, $\frac{216}{16} = \frac{216 \div 8}{16 \div 8} = \frac{27}{2}$. Now our expression looks like $\sqrt[12]{\frac{27}{2}}$.

We're getting closer! Now we have a simplified expression under a single 12th root. Let's see how this compares to our options.

Evaluating the Options

Now, let's examine the options provided and see which one matches our simplified expression, $\sqrt[12]{\frac{27}{2}}$. Remember, we're looking for an expression that is equivalent, meaning it has the same value.

Option 1: $\frac{\sqrt[12]{27}}{2}$

This option looks interesting because it has a 12th root in it, just like our simplified expression. However, the 2 in the denominator isn't under the root. To compare this with our expression, we need to rewrite it so that everything is under the 12th root. We can rewrite 2 as $\sqrt[12]{2^{12}}$, so the expression becomes $\frac{\sqrt[12]{27}}{\sqrt[12]{2^{12}}}$. Combining the roots, we get $\sqrt[12]{\frac{27}{2^{12}}}$, which is not the same as $\sqrt[12]{\frac{27}{2}}$. So, this option is not equivalent.

Option 2: $\frac{\sqrt[4]{24}}{2}$

This option has a 4th root, which is different from the 12th root we have in our simplified expression. To compare them, we would need to convert the 4th root to a 12th root. To do this, we can raise the expression inside the root to the power of 3 (since $4 \times 3 = 12$) and multiply the denominator by an appropriate factor. Let's rewrite this as $\frac{24^{\frac{1}{4}}}{2}$. To get a 12th root, we raise 24 to the power of $\frac{3}{12}$, so we have $\frac{(24^3)^{\frac{1}{12}}}{2}$, which simplifies to $\frac{\sqrt[12]{24^3}}{2}$. This is $\frac{\sqrt[12]{13824}}{2}$. Rewriting 2 as $\sqrt[12]{2^{12}}$, we get $\frac{\sqrt[12]{13824}}{\sqrt[12]{4096}}$, which simplifies to $\sqrt[12]{\frac{13824}{4096}} = \sqrt[12]{\frac{27}{8}}$. This is also not equivalent to our simplified expression, $\sqrt[12]{\frac{27}{2}}$.

Option 3: $\frac{\sqrt[12]{55296}}{2}$

This option has a 12th root, which is promising. Let's see if we can manipulate it to match our simplified expression. We have $\frac{\sqrt[12]{55296}}{2}$. Again, let's rewrite 2 as $\sqrt[12]{2^{12}}$, so the expression becomes $\frac{\sqrt[12]{55296}}{\sqrt[12]{4096}}$. Combining the roots, we get $\sqrt[12]{\frac{55296}{4096}}$. Let's simplify the fraction $\frac{55296}{4096}$. Dividing both the numerator and the denominator by 2048, we get $\frac{27}{2}$. So, the expression simplifies to $\sqrt[12]{\frac{27}{2}}$. Bingo! This matches our simplified expression.

Option 4: $\frac{\sqrt[12]{177147}}{3}$

Finally, let's look at the last option. We have $\frac{\sqrt[12]{177147}}{3}$. Rewriting 3 as $\sqrt[12]{3^{12}}$, we get $\frac{\sqrt[12]{177147}}{\sqrt[12]{531441}}$. Combining the roots, we have $\sqrt[12]{\frac{177147}{531441}}$. Simplifying this fraction, we get $\sqrt[12]{\frac{1}{3}}$, which is clearly not equivalent to $\sqrt[12]{\frac{27}{2}}$.

The Grand Finale: The Equivalent Expression

After carefully evaluating all the options, we've found that Option 3, $\frac{\sqrt[12]{55296}}{2}$, is the expression equivalent to $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$. Woohoo! We did it!

Steps to the Solution

To recap, here's what we did to solve this problem:

1. Rewrote the roots as fractional exponents: $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$ became $\frac{6^{\frac{1}{4}}}{2^{\frac{1}{3}}}$.
2. Found a common denominator for the exponents: We rewrote the exponents with a denominator of 12, giving us $\frac{6^{\frac{3}{12}}}{2^{\frac{4}{12}}}$.
3. Combined the terms under a single root: We got $\frac{\sqrt[12]{6^3}}{\sqrt[12]{2^4}}$, which simplified to $\frac{\sqrt[12]{216}}{\sqrt[12]{16}}$.
4. Simplified the fraction under the root: We obtained $\sqrt[12]{\frac{216}{16}}$, which simplified to $\sqrt[12]{\frac{27}{2}}$.
5. Compared the simplified expression with the options: We found that $\frac{\sqrt[12]{55296}}{2}$ is equivalent to $\sqrt[12]{\frac{27}{2}}$.

Why This Matters

This problem demonstrates the power of understanding and applying the rules of exponents and roots. By converting roots to fractional exponents, finding common denominators, and simplifying fractions, we were able to transform a complex-looking expression into a manageable form and find its equivalent. These skills are invaluable in mathematics and various scientific fields.

Final Thoughts

So, there you have it, guys! We successfully navigated this mathematical maze and found the equivalent expression. Remember, the key to solving these types of problems is to break them down into smaller, more manageable steps. Don't be intimidated by complex expressions; with a little bit of know-how and some practice, you can conquer any math challenge that comes your way. Keep exploring, keep learning, and most importantly, keep having fun with math!