Rationalizing Denominator Of Fourth Root Expression: A Step-by-Step Guide
Hey guys! Let's dive into this math problem together. We're going to tackle rationalizing the denominator of the expression $\sqrt[4]{\frac{81}{4 x^7}}$. This might sound intimidating, but don't worry, we'll break it down step by step. We'll make sure to explain everything clearly so you can follow along easily. So, grab your pencils and let's get started!
Understanding the Problem
So, the core of our mission is to rationalize the denominator of $\sqrt[4]{\frac{81}{4 x^7}}$. What does that even mean? Well, in math lingo, rationalizing the denominator means getting rid of any pesky radicals (like square roots, cube roots, and in our case, fourth roots) from the bottom of a fraction. Think of it as making the denominator a nice, clean, rational number – no more radicals allowed!
In our specific problem, we have a fourth root in the denominator, which means we need to transform the expression so that the denominator doesn't have any fourth roots. We're given that all variables represent positive real numbers, which is super helpful because it means we don't have to worry about any tricky situations with negative numbers under the radical. This simplifies our work and lets us focus on the core task: banishing that fourth root from the denominator.
The expression we're dealing with is $\sqrt[4]{\frac{81}{4 x^7}}$. The first thing we might notice is that we have a fraction inside the fourth root. A clever way to handle this is to remember that the fourth root of a fraction is the same as the fourth root of the numerator divided by the fourth root of the denominator. This means we can rewrite our expression as $\frac{\sqrt[4]{81}}{\sqrt[4]{4 x^7}}$. This separation is a key step in rationalizing the denominator because it allows us to focus on the numerator and denominator individually.
Now, let’s look at each part more closely. The numerator is $\sqrt[4]{81}$. Can we simplify this? Absolutely! We need to think, “What number, when raised to the fourth power, equals 81?” Well, $3^4 = 3 \times 3 \times 3 \times 3 = 81$, so $\sqrt[4]{81} = 3$. That simplifies our numerator nicely. The denominator, $\sqrt[4]{4 x^7}}$, is a bit more complex, and that’s where most of our work will be focused.
To rationalize this denominator, we need to figure out what we can multiply it by to get rid of the fourth root. Remember, we want the expression inside the fourth root to become a perfect fourth power. This means every factor inside the root should have an exponent that is a multiple of 4. Currently, we have $4 x^7$ inside the root. We can think of 4 as $2^2$, so we have $2^2 x^7$. To make the exponents multiples of 4, we need to multiply by something that will give us $2^4$ and $x^8$. This will help us determine what to multiply both the numerator and the denominator by to rationalize the denominator completely.
Breaking Down the Radicals
Let's dive deeper into breaking down the radicals. As we mentioned earlier, we have the expression $\sqrt[4]\frac{81}{4 x^7}}$. The crucial first step is to separate the radical using the property that the fourth root of a fraction is the fraction of the fourth roots. This gives us}{\sqrt[4]{4 x^7}}$
This separation is super helpful because it allows us to deal with the numerator and the denominator independently. Let’s start with the numerator, $\sqrt[4]{81}$. We need to find a number that, when raised to the fourth power, equals 81. If you think about it, $3^4 = 3 \times 3 \times 3 \times 3 = 81$. So, $\sqrt[4]{81} = 3$. The numerator simplifies beautifully to just 3, which makes our expression:
Now, let's tackle the denominator, $\sqrt[4]{4 x^7}}$. This is where the real work of rationalizing the denominator comes in. We need to figure out how to get rid of this fourth root. To do that, we need to make the expression inside the fourth root a perfect fourth power. A perfect fourth power is an expression where every factor has an exponent that is a multiple of 4.
Currently, we have $4x^7$ inside the fourth root. We can rewrite 4 as $2^2$, so we really have $2^2 x^7$. Think of it like this: we have two factors of 2 and seven factors of x. To make these perfect fourth powers, we need the exponents to be multiples of 4. For the 2, we need two more factors of 2 to get $2^4$. For the x, we need one more factor of x to get $x^8$.
So, we need to multiply the expression inside the fourth root by $2^2 x^1$, which is $4x$. This will give us $2^4 x^8$ inside the fourth root. But remember, whatever we multiply inside the root, we need to do it in a way that doesn't change the overall value of the fraction. This means we’ll need to multiply both the numerator and the denominator by a carefully chosen expression that will help us rationalize the denominator.
Rationalizing the Denominator
Okay, now for the exciting part: actually rationalizing the denominator! We’ve simplified our expression to $\frac{3}{\sqrt[4]{4 x^7}}$, and we've figured out that we need to transform the denominator so that the expression inside the fourth root becomes a perfect fourth power. Remember, we identified that we need to multiply $4x^7$ by $4x$ to achieve this.
So, we're going to multiply both the numerator and the denominator by $\sqrt[4]{4x}$. Why $\sqrt[4]{4x}$? Because this is the fourth root that, when multiplied by $\sqrt[4]{4x^7}$, will give us a perfect fourth power inside the radical. Here's how it looks:
When we multiply the numerators, we get $3\sqrt[4]{4x}$. For the denominators, we have $\sqrt[4]{4 x^7} \times \sqrt[4]{4x}$. Using the property that $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$, we can combine these under one radical:
Now, let's simplify $\sqrt[4]{16 x^8}$. We know that $16 = 2^4$, so we have $\sqrt[4]{2^4 x^8}$. The fourth root of $2^4$ is 2, and the fourth root of $x^8$ is $x^2$ (since $(x2)4 = x^8$). So, the denominator simplifies to $2x^2$. Our expression now looks like this:
And there we have it! We've successfully rationalized the denominator. The denominator is now $2x^2$, which is a rational expression—no more fourth roots in sight. The final simplified expression is $\frac{3\sqrt[4]{4x}}{2x^2}$.
Final Solution
Alright, let's wrap this up and make sure we've got the final answer crystal clear. We started with the expression $\sqrt[4]{\frac{81}{4 x^7}}$ and our mission was to rationalize the denominator. After breaking down the problem step by step, separating the radicals, identifying what we needed to multiply by, and doing the multiplication and simplification, we arrived at our final solution.
The original expression was:
We separated the radicals:
Simplified the numerator:
Rationalized the denominator by multiplying both the numerator and the denominator by $\sqrt[4]{4x}$:
Simplified the denominator:
So, our final, simplified expression with a rationalized denominator is:
This is the answer you'd put in the box. We've taken a somewhat complex-looking expression and transformed it into a more manageable form by getting rid of the radical in the denominator. Great job, guys! You've tackled a tricky problem and come out on top. Keep practicing, and you'll become a pro at rationalizing denominators in no time!
Rationalizing the denominator involves eliminating any radicals from the denominator of a fraction. In this case, we're dealing with the expression $\sqrt[4]{\frac{81}{4 x^7}}$, and our goal is to rewrite it without any fourth roots in the denominator. This process is crucial in mathematics because it simplifies expressions, making them easier to work with in further calculations and analyses. The given expression contains a fourth root, which means we need to manipulate it so that the denominator becomes a rational number. This often involves multiplying both the numerator and the denominator by a suitable expression that will eliminate the radical in the denominator. Understanding the steps involved in rationalizing the denominator not only helps in solving specific problems but also enhances overall algebraic skills and the ability to manipulate complex expressions. In this detailed explanation, we will break down each step, providing a clear and comprehensive approach to solving this problem. First, recognize that the expression involves a fourth root and a fraction. The key strategy here is to separate the radical over the fraction, simplify the numerator if possible, and then focus on eliminating the radical in the denominator. This requires a solid understanding of radical properties and exponent rules. The ultimate aim is to transform the denominator into a form where the radicand (the expression inside the radical) becomes a perfect fourth power. This means that each factor within the radical must have an exponent that is a multiple of 4. To achieve this, we'll identify the missing factors and multiply both the numerator and denominator by an appropriate expression. This methodical approach will not only solve the problem at hand but also equip you with the necessary skills to tackle similar problems in the future. So, let's dive in and break down the process step-by-step to make it easy to understand and apply.
Breaking Down the Expression
To effectively rationalize the denominator, the initial step involves simplifying the given expression and understanding its components. The expression we are working with is $\sqrt[4]\frac{81}{4 x^7}}$. First, recognize that the fourth root applies to the entire fraction. A crucial property of radicals is that the nth root of a fraction can be separated into the nth root of the numerator divided by the nth root of the denominator. Mathematically, this is expressed asb}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$. Applying this property to our expression, we can rewrite it as}\sqrt[4]{4 x^7}}$. This separation is a significant step because it allows us to deal with the numerator and the denominator independently, simplifying the process. Now, let's focus on the numerator, which is $\sqrt[4]{81}$. We need to determine if 81 is a perfect fourth power, meaning whether there is an integer that, when raised to the fourth power, equals 81. Recall that a perfect fourth power is a number that can be written as $n^4$ for some integer n. By recognizing that $81 = 3^4$, we can simplify the numerator = \sqrt[4]3^4} = 3$. This simplifies our expression to{\sqrt[4]{4 x^7}}$. Next, we turn our attention to the denominator, $\sqrt[4]{4 x^7}}$. This is where the bulk of the work lies in rationalizing the denominator. The goal is to eliminate the fourth root from the denominator, which means we need to transform the expression inside the radical into a perfect fourth power. To achieve this, we first express 4 as a power of its prime factor, which is $4 = 2^2$. So, the denominator becomes $\sqrt[4]{2^2 x^7}}$. Understanding the composition of the denominator is essential for determining what we need to multiply by to make the radicand a perfect fourth power. Each factor inside the radical needs to have an exponent that is a multiple of 4. In the next section, we will discuss how to identify the missing factors and construct the appropriate expression to multiply by.
Identifying the Missing Factors
After simplifying the expression to $\frac{3}{\sqrt[4]{2^2 x^7}}}$, the next crucial step is to identify the missing factors needed to make the radicand (the expression inside the radical) a perfect fourth power. To effectively rationalize the denominator, we need to transform the expression inside the fourth root so that each factor has an exponent that is a multiple of 4. This is because the fourth root of a perfect fourth power is a rational number, allowing us to eliminate the radical from the denominator. Currently, in the denominator, we have $\sqrt[4]{2^2 x^7}}$. The radicand is $2^2 x^7$. To determine what factors are missing, we look at the exponents of each factor. For the factor 2, the exponent is 2. To make this a multiple of 4, we need to increase it to 4. Thus, we need two more factors of 2, which can be expressed as $2^2$. For the factor x, the exponent is 7. The next multiple of 4 greater than 7 is 8. Therefore, we need one more factor of x, which is $x^1$ (or simply x). Combining these missing factors, we need to multiply the radicand by $2^2 x^1$, which simplifies to $4x$. However, we can't simply multiply the radicand by this factor; we must multiply the entire denominator by the fourth root of this factor. So, we will multiply the denominator by $\sqrt[4]{4x}}$. To maintain the equality of the expression, we must also multiply the numerator by the same factor. This is a critical step in rationalizing the denominator, as it ensures that we are only changing the form of the expression, not its value. Therefore, we multiply both the numerator and the denominator by $\sqrt[4]{4x}}$. By identifying these missing factors, we set the stage for the next step, which involves multiplying both the numerator and the denominator by the appropriate expression to eliminate the radical from the denominator. In the subsequent section, we will carry out this multiplication and simplify the resulting expression.
Multiplying and Simplifying
Having identified the missing factors, we now proceed to multiply both the numerator and the denominator by the appropriate expression. Our expression is currently in the form $\frac3}{\sqrt[4]{2^2 x^7}}}$, and we determined that we need to multiply by $\sqrt[4]{4x}}$ to rationalize the denominator. Multiplying both the numerator and the denominator by $\sqrt[4]{4x}}$, we get\sqrt[4]{2^2 x^7}} \times \frac{\sqrt[4]{4x}}{\sqrt[4]{4x}}$. First, let's multiply the numerators = 3\sqrt[4]4x}$. This simply combines the integer 3 with the fourth root term, resulting in $3\sqrt[4]{4x}$. Next, we multiply the denominators \times \sqrt[4]4x}$. Using the property that $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$, we can combine these under one radical$. Now, we simplify the expression inside the fourth root: $2^2 x^7 \times 4x = 2^2 x^7 \times 2^2 x = 2^2+2} x^{7+1} = 2^4 x^8$. So, the denominator becomes $\sqrt[4]{2^4 x^8}$. Now, we simplify the fourth root$. Since both exponents are multiples of 4, we can take the fourth root of each factor: $\sqrt[4]2^4} = 2$ and $\sqrt[4]{x^8} = x^{8/4} = x^2$. Therefore, the simplified denominator is $2x^2$. Our expression now looks like}{2x^2}$. Finally, we check if there are any further simplifications. In this case, the expression is fully simplified, and we have successfully rationalized the denominator. The denominator $2x^2$ contains no radicals. In the next section, we will present the final answer and summarize the steps taken to solve the problem.
Presenting the Final Answer
After meticulously working through the steps, we have successfully rationalized the denominator of the given expression. To recap, we started with: $\sqrt[4]\frac{81}{4 x^7}}$. We first separated the radical over the fraction}\sqrt[4]{4 x^7}}$. Then, we simplified the numerator\sqrt[4]{4 x^7}}$. Next, we identified the missing factors needed to make the radicand in the denominator a perfect fourth power, which led us to multiply both the numerator and the denominator by $\sqrt[4]{4x}}$\sqrt[4]{4 x^7}} \times \frac{\sqrt[4]{4x}}{\sqrt[4]{4x}}$. We then multiplied the numerators and denominators}\sqrt[4]{4 x^7 \times 4x}} = \frac{3\sqrt[4]{4x}}{\sqrt[4]{16 x^8}}$. Finally, we simplified the denominator}2x^2}$. Therefore, the final answer, with the denominator rationalized, is}{2x^2}$. This is the simplified form of the original expression, where the denominator no longer contains any radicals. This process not only solves the problem but also provides a structured approach to tackling similar problems involving radicals and rationalization. In summary, rationalizing the denominator involves eliminating radicals from the denominator of a fraction, which often makes the expression easier to work with in subsequent calculations. The key steps include separating radicals, identifying missing factors, multiplying both the numerator and denominator by an appropriate expression, and simplifying the result. By following these steps, you can effectively rationalize denominators and simplify complex algebraic expressions.
How do I rationalize the denominator of the expression $\sqrt[4]{\frac{81}{4 x^7}}$, assuming all variables represent positive real numbers?
Rationalizing Denominator of Fourth Root Expression A Step-by-Step Guide