Simplifying Radical Expressions A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving deep into the world of radical expressions to simplify a seemingly complex problem. Our mission, should we choose to accept it, is to find the simplified form of the following expression, keeping in mind that $x ext{ and } y$ are non-negative (i.e., $x \geq 0$ and $y \geq 0$):

2(16x4)−2(2y4)+3(81x4)−4(32y4)2(\sqrt[4]{16 x})-2(\sqrt[4]{2 y})+3(\sqrt[4]{81 x})-4(\sqrt[4]{32 y})

Buckle up, because we're about to embark on a step-by-step journey to unravel this mathematical puzzle!

Breaking Down the Expression: A Step-by-Step Approach

Initial Assessment

Okay, guys, let's take a good look at what we've got. At first glance, this expression might seem a bit intimidating with all those fourth roots. But don't worry, we're going to break it down into manageable pieces. The key here is to identify terms that can be simplified by finding perfect fourth powers within the radicals. Remember, a perfect fourth power is a number that can be obtained by raising an integer to the fourth power (e.g., 1, 16, 81, 256, etc.). Our goal is to express the numbers inside the radicals as products of perfect fourth powers and other factors.

Simplifying the First Term: $2(\sqrt[4]{16 x})$

Let's start with the first term: $2(\sqrt[4]{16 x})$. Within this term, we have the fourth root of 16x. Can we simplify this further? Absolutely! We know that 16 is a perfect fourth power because $2^4 = 16$. So, we can rewrite the term as follows:

2(16x4)=2(164â‹…x4)2(\sqrt[4]{16 x}) = 2(\sqrt[4]{16} \cdot \sqrt[4]{x})

Now, we can take the fourth root of 16, which is 2:

2(164â‹…x4)=2(2x4)=4x42(\sqrt[4]{16} \cdot \sqrt[4]{x}) = 2(2 \sqrt[4]{x}) = 4\sqrt[4]{x}

So, the first term simplifies to $4\sqrt[4]{x}$. Not too shabby, right?

Tackling the Second Term: $-2(\sqrt[4]{2 y})$

Moving on to the second term: $-2(\sqrt[4]2 y})$. In this case, we have the fourth root of 2y. Is there a perfect fourth power hiding in there? Unfortunately, no. The number 2 is a prime number, and it's not a perfect fourth power. So, this term is already in its simplest form. We'll just carry it along as is $-2(\sqrt[4]{2 y)$.

Decoding the Third Term: $3(\sqrt[4]{81 x})$

Now, let's tackle the third term: $3(\sqrt[4]{81 x})$. Here, we have the fourth root of 81x. Do we spot a perfect fourth power? Yes, we do! The number 81 is a perfect fourth power because $3^4 = 81$. So, we can rewrite this term as:

3(81x4)=3(814â‹…x4)3(\sqrt[4]{81 x}) = 3(\sqrt[4]{81} \cdot \sqrt[4]{x})

Taking the fourth root of 81 gives us 3:

3(814â‹…x4)=3(3x4)=9x43(\sqrt[4]{81} \cdot \sqrt[4]{x}) = 3(3 \sqrt[4]{x}) = 9\sqrt[4]{x}

Thus, the third term simplifies to $9\sqrt[4]{x}$. We're on a roll!

Unraveling the Fourth Term: $-4(\sqrt[4]{32 y})$

Finally, let's conquer the fourth term: $-4(\sqrt[4]{32 y})$. This one looks a bit trickier, but we can handle it. We have the fourth root of 32y. Is 32 a perfect fourth power? Nope, but can we factor out a perfect fourth power from 32? Yes, we can! We know that $2^4 = 16$, and 16 is a factor of 32. In fact, $32 = 16 \cdot 2$. So, we can rewrite the term as:

−4(32y4)=−4(16⋅2y4)=−4(164⋅2y4)-4(\sqrt[4]{32 y}) = -4(\sqrt[4]{16 \cdot 2y}) = -4(\sqrt[4]{16} \cdot \sqrt[4]{2y})

Taking the fourth root of 16 gives us 2:

−4(164⋅2y4)=−4(22y4)=−82y4-4(\sqrt[4]{16} \cdot \sqrt[4]{2y}) = -4(2 \sqrt[4]{2y}) = -8\sqrt[4]{2y}

Therefore, the fourth term simplifies to $-8\sqrt[4]{2y}$.

Combining Like Terms: The Grand Finale

Now that we've simplified each term, let's put it all together:

4x4−22y4+9x4−82y44\sqrt[4]{x} - 2\sqrt[4]{2 y} + 9\sqrt[4]{x} - 8\sqrt[4]{2y}

To simplify further, we need to combine like terms. Remember, like terms are terms that have the same variable part, including the radical. In this expression, we have two types of terms: terms with $\sqrt[4]{x}$ and terms with $\sqrt[4]{2y}$.

Let's group the like terms together:

(4x4+9x4)+(−22y4−82y4)(4\sqrt[4]{x} + 9\sqrt[4]{x}) + (-2\sqrt[4]{2 y} - 8\sqrt[4]{2y})

Now, we can combine the coefficients of the like terms:

(4+9)x4+(−2−8)2y4(4 + 9)\sqrt[4]{x} + (-2 - 8)\sqrt[4]{2y}

13x4+(−10)2y413\sqrt[4]{x} + (-10)\sqrt[4]{2y}

Finally, we can rewrite the expression to make it look a bit cleaner:

13x4−102y413\sqrt[4]{x} - 10\sqrt[4]{2y}

The Simplified Form: Victory Achieved!

And there you have it, folks! The simplified form of the expression $2(\sqrt[4]{16 x})-2(\sqrt[4]{2 y})+3(\sqrt[4]{81 x})-4(\sqrt[4]{32 y})$ is:

13x4−102y4\boxed{13\sqrt[4]{x} - 10\sqrt[4]{2y}}

We did it! We successfully navigated the world of radical expressions and emerged victorious. Give yourselves a pat on the back!

Key Takeaways and Pro Tips

Before we wrap up, let's recap some of the key concepts and pro tips that helped us conquer this problem:

  • Identify Perfect Fourth Powers: The first step in simplifying radical expressions with fourth roots is to look for perfect fourth powers within the radicals. Remember, a perfect fourth power is a number that can be obtained by raising an integer to the fourth power (e.g., 1, 16, 81, 256, etc.).
  • Factor Out Perfect Fourth Powers: If you find a perfect fourth power as a factor within the radical, factor it out. This allows you to simplify the expression by taking the fourth root of the perfect fourth power.
  • Simplify Each Term Individually: Break down the expression into individual terms and simplify each term separately. This makes the problem more manageable and less overwhelming.
  • Combine Like Terms: After simplifying each term, look for like terms. Like terms have the same variable part, including the radical. Combine the coefficients of like terms to further simplify the expression.
  • Don't Be Afraid to Break It Down: Radical expressions can seem daunting at first, but by breaking them down into smaller, more manageable steps, you can conquer even the most complex problems.

Practice Makes Perfect

Like any mathematical skill, simplifying radical expressions requires practice. The more you practice, the more comfortable and confident you'll become. So, go ahead and try some more problems! You can find plenty of practice problems online or in your math textbook.

Final Thoughts

Simplifying radical expressions is an important skill in algebra and beyond. It allows you to express mathematical expressions in their most concise and understandable form. By mastering the techniques we've discussed today, you'll be well-equipped to tackle any radical expression that comes your way.

Keep practicing, keep exploring, and keep having fun with math!