Simplifying Radicals Which Expression Is Equivalent To $\frac{\sqrt{10}}{\sqrt[4]{8}}$

In this article, we will delve into the fascinating world of radical expressions, specifically focusing on simplifying and finding equivalent forms. Our main task is to determine which expression is equivalent to the given expression $\frac{\sqrt{10}}{\sqrt[4]{8}}$. This problem involves understanding the properties of radicals and exponents, as well as the process of rationalizing denominators. We will explore the steps to simplify the given expression and then compare it with the provided options to identify the correct equivalent expression. Let's embark on this mathematical journey together!

Understanding the Problem: Radicals and Exponents

To begin, it is crucial to grasp the fundamental concepts of radicals and exponents. A radical is a mathematical expression that involves a root, such as a square root, cube root, or fourth root. The general form of a radical is $\sqrt[n]{a}$, where n is the index (the type of root) and a is the radicand (the value inside the root). For instance, in the expression $\sqrt{10}$, the index is 2 (since it's a square root) and the radicand is 10. Similarly, in $\sqrt[4]{8}$, the index is 4 (a fourth root) and the radicand is 8.

Exponents provide a shorthand notation for repeated multiplication. For example, $2^3$ means 2 multiplied by itself three times (2 * 2 * 2 = 8). Fractional exponents are closely related to radicals. The expression $a^{\frac{1}{n}}$ is equivalent to $\sqrt[n]{a}$. This relationship is key to simplifying radical expressions, as it allows us to convert between radical and exponential forms, leveraging the rules of exponents.

When dealing with radical expressions, several properties come into play. One crucial property is the product rule, which states that the nth root of a product is the product of the nth roots: $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Another important property is the quotient rule, which states that the nth root of a quotient is the quotient of the nth roots: $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$. These properties allow us to break down complex radicals into simpler forms. To effectively simplify $\frac{\sqrt{10}}{\sqrt[4]{8}}$, we will employ these principles to express both the numerator and the denominator in terms of common exponents, thereby facilitating the simplification process. Recognizing that 8 can be written as $2^3$, we can rewrite $\sqrt[4]{8}$ as $(2^3)^{\frac{1}{4}} = 2^{\frac{3}{4}}$. This conversion to exponential form will aid in further simplification and comparison with the given options. The properties of exponents, such as the power of a power rule ((a^m)n = a^{mn}), are essential tools in manipulating these expressions. By understanding these foundational concepts, we can approach the problem methodically and arrive at the correct equivalent expression.

Step-by-Step Simplification of $\frac{\sqrt{10}}{\sqrt[4]{8}}$

Now, let's dive into the step-by-step simplification of the given expression $\frac{\sqrt{10}}{\sqrt[4]{8}}$. Our goal is to manipulate this expression into a form that matches one of the provided options. This process will involve converting radicals to exponential forms, applying exponent rules, and rationalizing the denominator if necessary.

Step 1: Convert Radicals to Exponential Forms

The first step is to convert both the numerator and the denominator from radical form to exponential form. Recall that $\sqrt[n]{a} = a^{\frac{1}{n}}$.

• $\sqrt{10}$ can be written as $10^{\frac{1}{2}}$.
• $\sqrt[4]{8}$ can be written as $8^{\frac{1}{4}}$.

So, the expression becomes $\frac{10^{\frac{1}{2}}}{8^{\frac{1}{4}}}$. This conversion allows us to work with exponents, which are often easier to manipulate than radicals. The fractional exponents provide a clear way to combine and simplify terms.

Step 2: Express the Base in Prime Factors

Next, we express the bases (10 and 8) in terms of their prime factors. This will help us simplify the expression further.

• $10 = 2 \cdot 5$, so $10^{\frac{1}{2}} = (2 \cdot 5)^{\frac{1}{2}} = 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}$
• $8 = 2^3$, so $8^{\frac{1}{4}} = (2^3)^{\frac{1}{4}} = 2^{\frac{3}{4}}$

Now, the expression is $\frac{2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}}{2^{\frac{3}{4}}}$. Expressing the numbers in terms of prime factors allows us to apply exponent rules more effectively. The decomposition of 10 into 2 * 5 and 8 into $2^3$ is a crucial step in this process.

Step 3: Apply Exponent Rules to Simplify

To simplify the expression, we apply the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$.

• Divide $2^{\frac{1}{2}}$ by $2^{\frac{3}{4}}$: $\frac{2^{\frac{1}{2}}}{2^{\frac{3}{4}}} = 2^{\frac{1}{2} - \frac{3}{4}} = 2^{\frac{2}{4} - \frac{3}{4}} = 2^{-\frac{1}{4}}$

So, the expression becomes $2^{-\frac{1}{4}} \cdot 5^{\frac{1}{2}}$.

Step 4: Convert Back to Radical Form

Convert the expression back to radical form to make it easier to compare with the given options.

• $2^{-\frac{1}{4}} = \frac{1}{2^{\frac{1}{4}}} = \frac{1}{\sqrt[4]{2}}$
• $5^{\frac{1}{2}} = \sqrt{5}$

Thus, the expression is $\frac{\sqrt{5}}{\sqrt[4]{2}}$.

Step 5: Rationalize the Denominator

To rationalize the denominator, we need to eliminate the radical in the denominator. To do this, we multiply both the numerator and the denominator by a term that will make the denominator a perfect fourth power.

• Multiply both numerator and denominator by $\sqrt[4]{2^3}$ (which is $\sqrt[4]{8}$): $\frac{\sqrt{5}}{\sqrt[4]{2}} \cdot \frac{\sqrt[4]{2^3}}{\sqrt[4]{2^3}} = \frac{\sqrt{5} \cdot \sqrt[4]{8}}{\sqrt[4]{2} \cdot \sqrt[4]{8}} = \frac{\sqrt{5} \cdot \sqrt[4]{8}}{\sqrt[4]{16}}$

Since $\sqrt[4]{16} = 2$, the expression simplifies to $\frac{\sqrt{5} \cdot \sqrt[4]{8}}{2}$.

Step 6: Combine Radicals

We can express $\sqrt{5}$ as $\sqrt[4]{5^2} = \sqrt[4]{25}$. Now, multiply the radicals in the numerator:

• $\frac{\sqrt[4]{25} \cdot \sqrt[4]{8}}{2} = \frac{\sqrt[4]{25 \cdot 8}}{2} = \frac{\sqrt[4]{200}}{2}$

Therefore, the simplified expression is $\frac{\sqrt[4]{200}}{2}$. By carefully applying the rules of exponents and radicals, we have successfully simplified the given expression into an equivalent form. Each step, from converting to exponential form to rationalizing the denominator, was crucial in reaching the final answer.

Comparing the Simplified Expression with the Options

After simplifying the given expression $\frac{\sqrt{10}}{\sqrt[4]{8}}$, we arrived at the equivalent expression $\frac{\sqrt[4]{200}}{2}$. Now, we need to compare this simplified expression with the options provided to identify the correct answer.

The options are:

A. $\frac{\sqrt[4]{200}}{2}$ B. $\frac{\sqrt[4]{20}}{2}$ C. $\frac{2 \sqrt{5}}{5}$ D. $\frac{100}{8}$

By direct comparison, we can see that our simplified expression $\frac{\sqrt[4]{200}}{2}$ matches option A exactly. Therefore, option A is the correct equivalent expression.

Why Other Options Are Incorrect

To further solidify our understanding, let's briefly examine why the other options are incorrect.

• Option B: $\frac{\sqrt[4]{20}}{2}$ This option is close to the correct answer but has a different radicand. We simplified the original expression to $\frac{\sqrt[4]{200}}{2}$, not $\frac{\sqrt[4]{20}}{2}$. The difference arises from the simplification and rationalization steps, where the radicand was correctly calculated as 200, not 20.

• Option C: $\frac{2 \sqrt{5}}{5}$ This option is in a different form altogether. It involves a square root and a fraction with 5 in the denominator. While it might seem like a possible simplification, our step-by-step simplification process clearly led us to an expression with a fourth root in the numerator and 2 in the denominator. To convert our simplified expression to this form would require additional, incorrect steps.

• Option D: $\frac{100}{8}$ This option is a simple fraction and does not involve any radicals. It is highly unlikely that a radical expression would simplify to a simple fraction without radicals. This option is easily ruled out due to its form, which is drastically different from the radical expressions we are dealing with. Each incorrect option highlights the importance of following the correct simplification steps. Our methodical approach, from converting radicals to exponential forms to rationalizing the denominator, ensures that we arrive at the correct equivalent expression. This comparative analysis not only confirms the correctness of option A but also reinforces our understanding of why the other options are not equivalent to the original expression. The precision in each step is crucial, and a minor error can lead to a completely different result. By understanding these nuances, we can confidently tackle similar problems in the future.

Conclusion: The Equivalent Expression

In conclusion, after a detailed step-by-step simplification process, we have determined that the expression equivalent to $\frac{\sqrt{10}}{\sqrt[4]{8}}$ is A. $\frac{\sqrt[4]{200}}{2}$. This solution was obtained by converting radicals to exponential forms, simplifying using exponent rules, and rationalizing the denominator.

We started by understanding the fundamental concepts of radicals and exponents, which are the building blocks of this problem. The ability to convert between radical and exponential forms, along with a firm grasp of exponent rules, allowed us to manipulate the given expression effectively. Expressing the bases in terms of their prime factors was a key step in simplifying the radicals.

The process of rationalizing the denominator was crucial in transforming the expression into a form that could be easily compared with the given options. By multiplying both the numerator and the denominator by an appropriate radical, we eliminated the radical from the denominator, leading us closer to the final simplified form.

Comparing our simplified expression with the provided options, we found a direct match with option A, $\frac{\sqrt[4]{200}}{2}$. We also analyzed why the other options were incorrect, reinforcing our understanding of the simplification process. This analysis highlighted the importance of each step and the precision required in manipulating radical expressions. Throughout the simplification, we emphasized the importance of understanding each step, from the initial conversion to exponential form to the final rationalization. The systematic approach ensures that we avoid common pitfalls and arrive at the correct answer. By mastering these techniques, we can confidently tackle similar problems involving radical expressions.

This problem serves as an excellent example of how a methodical approach, combined with a solid understanding of fundamental concepts, can lead to the successful simplification of complex expressions. The skills and techniques learned in this exercise are valuable tools in the broader field of mathematics, particularly in algebra and calculus, where radical expressions frequently appear. Understanding these principles not only helps in solving specific problems but also enhances the overall mathematical problem-solving ability. The ability to manipulate and simplify expressions is a core skill that extends beyond specific mathematical contexts, fostering logical thinking and precision in problem-solving.

In summary, the journey from the initial expression $\frac{\sqrt{10}}{\sqrt[4]{8}}$ to the equivalent form $\frac{\sqrt[4]{200}}{2}$ showcases the power of mathematical simplification. The careful application of exponent rules, radical properties, and rationalization techniques transforms a seemingly complex expression into a more manageable and understandable form. This process not only provides the correct answer but also deepens our understanding of the underlying mathematical principles.