Equivalent Expression Of √10/√8: Step-by-Step Solution

Hey guys! Let's break down this math problem together. The question asks: What expression is equivalent to $\frac{\sqrt{10}}{\sqrt[4]{8}}$? This might seem tricky at first, but don't worry, we'll go through it step by step. We're going to explore how to simplify radical expressions and identify equivalent forms. It's all about manipulating those roots and exponents to get to the right answer. So, grab your pencils, and let's dive in!

Understanding the Problem

Before we jump into solving, let's make sure we understand exactly what the question is asking. The core of the problem lies in simplifying the expression $\frac{\sqrt{10}}{\sqrt[4]{8}}$. This involves dealing with square roots and fourth roots, which are just fractional exponents in disguise. Remember, $\sqrt{x}$ is the same as $x^{\frac{1}{2}}$, and $\sqrt[4]{x}$ is the same as $x^{\frac{1}{4}}$. The goal is to manipulate this expression using the rules of exponents and radicals to see which of the answer choices (A, B, C, or D) matches our simplified form.

Think of it like this: we're given a fraction with radicals in both the numerator and the denominator. To simplify, we need to find a common ground, which often means expressing everything in terms of the same root or exponent. This might involve rationalizing the denominator, combining terms, or simplifying individual radicals. By the end, we should have a cleaner, more manageable expression that we can easily compare to the given options. This kind of problem is common in algebra and pre-calculus, so mastering these techniques is super valuable!

Breaking Down the Radicals

Okay, let's start by breaking down the radicals individually. This is a crucial step in simplifying complex expressions. We have $\sqrt{10}$ in the numerator and $\sqrt[4]{8}$ in the denominator. Let's tackle them one at a time.

Simplifying the Numerator: √10

First up, we have $\sqrt{10}$. Can we simplify this further? Well, we need to see if 10 has any perfect square factors. The factors of 10 are 1, 2, 5, and 10. None of these (other than 1) are perfect squares (like 4, 9, 16, etc.). This means $\sqrt{10}$ is already in its simplest form. We can't break it down any further. So, for now, we'll leave it as $\sqrt{10}$. Sometimes, the simplest steps are the most important – recognizing when you can't simplify something is just as key as knowing how to simplify!

Simplifying the Denominator: √[4]8

Now, let's focus on the denominator: $\sqrt[4]{8}$. This is a fourth root, meaning we're looking for factors that appear four times. It might not be immediately obvious, but we can rewrite 8 as $2^3$. So, we have $\sqrt[4]{2^3}$. To make this easier to work with, let's convert it to exponential form. Remember, $\sqrt[n]{x^m}$ is the same as $x^{\frac{m}{n}}$. In our case, this means $\sqrt[4]{2^3}$ becomes $2^{\frac{3}{4}}$. This exponential form will be super helpful when we combine it with the numerator. By converting radicals to exponents, we can use the rules of exponents to simplify the whole expression more effectively.

Combining and Simplifying the Expression

Alright, we've simplified the numerator and denominator separately. Now it's time to put them back together and see what we can do. We started with $\frac{\sqrt{10}}{\sqrt[4]{8}}$. We know $\sqrt{10}$ is already in its simplest form, and we've rewritten $\sqrt[4]{8}$ as $2^{\frac{3}{4}}$. So, our expression now looks like this: $\frac{\sqrt{10}}{2^{\frac{3}{4}}}$.

This is where things get interesting. To combine these terms, it's helpful to express everything with the same type of exponent or root. Let's convert the numerator, $\sqrt{10}$, to exponential form as well. We know $\sqrt{10}$ is the same as $10^{\frac{1}{2}}$. Now our expression is: $\frac{10^{\frac{1}{2}}}{2^{\frac{3}{4}}}$. To make the exponents easier to compare, we need a common denominator for the fractions $\frac{1}{2}$ and $\frac{3}{4}$. The least common denominator is 4. So, let's rewrite $\frac{1}{2}$ as $\frac{2}{4}$. Our expression now becomes $\frac{10^{\frac{2}{4}}}{2^{\frac{3}{4}}}$.

But we are not done yet. We have a tricky fraction with exponents. To simplify this further, we need to get creative. One common strategy is to rationalize the denominator or rewrite the base of the numerator to match the base of the denominator if possible. Let's try manipulating the numerator to include a power of 2.

Rationalizing and Final Simplification

Okay, let's think about how to simplify $\frac{10^{\frac{2}{4}}}{2^{\frac{3}{4}}}$ further. The key here is to get rid of the fractional exponent in the denominator or to find a common base. Since 10 can be written as 2 * 5, let's try rewriting the numerator using this fact. We have $10^{\frac{2}{4}}$, which is the same as $(2 \cdot 5)^{\frac{2}{4}}$. Using the power of a product rule, we can rewrite this as $2^{\frac{2}{4}} \cdot 5^{\frac{2}{4}}$. Now our expression looks like this: $\frac{2^{\frac{2}{4}} \cdot 5^{\frac{2}{4}}}{2^{\frac{3}{4}}}$.

Now we have a power of 2 in both the numerator and the denominator, which is great! We can use the quotient rule for exponents, which says that $\frac{x^a}{x^b} = x^{a-b}$. So, $\frac{2^{\frac{2}{4}}}{2^{\frac{3}{4}}} = 2^{\frac{2}{4} - \frac{3}{4}} = 2^{-\frac{1}{4}}$. Putting it all together, our expression is now $2^{-\frac{1}{4}} \cdot 5^{\frac{2}{4}}$. Remember that a negative exponent means we take the reciprocal, so $2^{-\frac{1}{4}} = \frac{1}{2^{\frac{1}{4}}}$. And we can simplify $5^{\frac{2}{4}}$ to $5^{\frac{1}{2}}$, which is just $\sqrt{5}$. So our expression becomes $\frac{\sqrt{5}}{2^{\frac{1}{4}}}$.

Now, let's get rid of the fractional exponent in the denominator by multiplying both the numerator and denominator by $2^{\frac{3}{4}}$. This gives us $\frac{\sqrt{5} \cdot 2^{\frac{3}{4}}}{2^{\frac{1}{4}} \cdot 2^{\frac{3}{4}}} = \frac{\sqrt{5} \cdot 2^{\frac{3}{4}}}{2}$.

To match one of the answer choices, let's rewrite $2^{\frac{3}{4}}$ as a fourth root: $2^{\frac{3}{4}} = \sqrt[4]{2^3} = \sqrt[4]{8}$. So our expression is now $\frac{\sqrt{5} \cdot \sqrt[4]{8}}{2}$. We can combine the radicals by expressing \sqrt5} as a fourth root $\sqrt{5 = 5^{\frac{1}{2}} = 5^{\frac{2}{4}} = \sqrt[4]{5^2} = \sqrt[4]{25}$. Now we have $\frac{\sqrt[4]{25} \cdot \sqrt[4]{8}}{2}$. Multiplying the radicals gives us $\frac{\sqrt[4]{25 \cdot 8}}{2} = \frac{\sqrt[4]{200}}{2}$.

Conclusion

Wow, we made it! After all that simplification, we found that the expression $\frac{\sqrt{10}}{\sqrt[4]{8}}$ is equivalent to $\frac{\sqrt[4]{200}}{2}$. This matches answer choice A. Remember, the key to these problems is breaking them down step by step, using the rules of exponents and radicals, and not being afraid to manipulate the expressions until you find a match. You got this!