Equivalent Expression Of √(10) / ∜(8): Math Guide

Nov 1, 2025 by SLV Team 50 views

Hey guys! Let's dive into a fun math problem today where we'll figure out which expression is equivalent to the fraction √(10) / ∜(8). This kind of problem is super common in algebra and it's all about manipulating radicals and exponents to get to the right answer. We'll break it down step by step, so it’s easy to follow along, even if you're just starting to get the hang of radicals. So, grab your pencils, and let's get started!

Understanding the Problem

Okay, so our mission is to find an expression that looks different but has the same value as √(10) / ∜(8). To do this, we need to understand what square roots and fourth roots are, and how we can play around with them. Remember, a square root (√) is the same as raising something to the power of 1/2, and a fourth root (∜) is like raising something to the power of 1/4. This is a critical concept because it allows us to use the rules of exponents to simplify things. Let's keep this in mind as we move forward, because it’s going to be our secret weapon in solving this problem. It's like having a translator that lets us switch between radical language and exponent language, making the whole process much smoother and easier to grasp. With this understanding, we can start thinking about how to rewrite our original expression in a way that makes it easier to work with. Think of it as preparing the ingredients before we start cooking – we need to get everything in the right form before we can combine them effectively.

Initial Expression: √(10) / ∜(8)

Let's take a closer look at our initial expression: √(10) / ∜(8). The numerator is the square root of 10, and the denominator is the fourth root of 8. At first glance, it might not be obvious how to simplify this, but that's where our understanding of exponents comes in handy. We can rewrite the square root of 10 as 10^(1/2) and the fourth root of 8 as 8^(1/4). This transformation is the key to unlocking the problem because it allows us to use exponent rules to combine and simplify these terms. Think of it as converting from one currency to another – once we're in the same "exponent currency," we can start doing calculations. It’s like having a common language that allows us to perform mathematical operations more easily. By expressing everything in terms of exponents, we can apply rules like the quotient rule for exponents (which states that a^m / a^n = a^(m-n)) to simplify the expression. This is a standard technique in algebra, and mastering it will help you tackle a wide range of problems involving radicals and exponents. So, remember, when you see radicals, think exponents – it’s often the first step towards simplification.

Rewriting with Exponents

Now, let's put our exponent knowledge to work. We can rewrite √(10) as 10^(1/2) and ∜(8) as 8^(1/4). This gives us a new way to look at the problem: 10^(1/2) / 8^(1/4). But we're not done yet! To really simplify this, we need to express both the base numbers (10 and 8) in terms of their prime factors. This is like breaking down a complex word into its individual letters – it helps us see the underlying structure. The prime factorization of 10 is 2 * 5, and the prime factorization of 8 is 2^3. So, we can rewrite our expression as (2 * 5)^(1/2) / (2³⁾(1/4). This might seem like we've made things more complicated, but trust me, it's a crucial step. By expressing everything in terms of prime factors, we can apply the power of a product rule, which states that (ab)^n = a^n * b^n, and the power of a power rule, which states that (a^m)n = a^(m*n). These rules will allow us to further simplify the expression and get closer to our final answer. It’s like having a set of tools that allow us to disassemble a complex machine into its individual parts, making it easier to understand and work with.

Converting Radicals to Exponential Form

So, we've transformed √(10) / ∜(8) into 10^(1/2) / 8^(1/4). This step is super important because it allows us to use the rules of exponents, which are much easier to manipulate than radicals. Think of it like switching from a complicated map to a GPS – the GPS (exponents) gives us a clearer route. Now, let's break down 10 and 8 into their prime factors. 10 can be written as 2 * 5, and 8 can be written as 2^3. This gives us (2 * 5)^(1/2) / (2³⁾(1/4). Remember, the goal here is to get everything in the same base so we can combine the exponents. This is like trying to compare apples and oranges – we need to convert them to the same unit (like fruit) before we can make a meaningful comparison. By expressing everything in terms of its prime factors, we're setting ourselves up for the next step, which is to apply the power rules of exponents. This is where things start to get really interesting, and we'll see how these seemingly small changes can lead to significant simplifications. It’s like planting seeds – each step we take is a seed that will grow into a solution.

Prime Factorization

Breaking down 10 and 8 into their prime factors is a key move. 10 becomes 2 * 5, and 8 becomes 2 * 2 * 2, which we can write as 2^3. Now our expression looks like this: (2 * 5)^(1/2) / (2³⁾(1/4). This might seem like a small step, but it's actually huge! It sets us up to use some powerful exponent rules that will make the simplification process much smoother. Think of prime factorization as finding the fundamental building blocks of a number. Just like you can build any structure out of basic Lego bricks, you can build any number out of its prime factors. This allows us to see the underlying structure of the numbers and manipulate them more easily. In this case, by expressing 10 and 8 in terms of their prime factors, we're able to apply the power of a product rule and the power of a power rule, which are essential for simplifying expressions with exponents. It’s like having a secret code that allows us to unlock the hidden potential of the expression.

Applying Exponent Rules

Time to bring out the exponent rulebook! We've got (2 * 5)^(1/2) / (2³⁾(1/4). First, let's apply the power of a product rule to the numerator: (2 * 5)^(1/2) = 2^(1/2) * 5^(1/2). Then, we'll use the power of a power rule in the denominator: (2³⁾(1/4) = 2^(3/4). This transforms our expression into (2^(1/2) * 5^(1/2)) / 2^(3/4). See how things are starting to simplify? Now we have the same base (2) in both the numerator and the denominator, which means we can use the quotient rule for exponents. This is like having all the ingredients for a recipe – now we can finally start cooking! By applying these exponent rules, we're essentially rearranging the pieces of the puzzle to make the solution clearer. Each rule we apply is like a step in a dance, moving us closer to the final pose. Remember, the key to mastering these kinds of problems is to practice and get comfortable with the different exponent rules. The more you use them, the more natural they will become, and the easier it will be to see how to apply them in different situations.

Power of a Product

Using the power of a product rule, we can break down (2 * 5)^(1/2) into 2^(1/2) * 5^(1/2). This rule basically says that if you have a product raised to a power, you can raise each factor to that power separately. It's like distributing the exponent to each term inside the parentheses. This is a really handy rule because it allows us to separate out the terms and deal with them individually. In our case, it allows us to isolate the square root of 2 and the square root of 5, which makes the expression easier to manage. Think of it as untangling a knot – by separating the strands, we can see how they're connected and make it easier to work with. This step is crucial because it sets us up for the next rule, which is the power of a power rule. By applying the power of a product rule, we're essentially preparing the ground for the next level of simplification. It’s like organizing your workspace before starting a project – by having everything in its place, you can work more efficiently and effectively.

Power of a Power

Next up is the power of a power rule. We have (2³⁾(1/4) in the denominator. This rule tells us that when we raise a power to another power, we multiply the exponents. So, (2³⁾(1/4) = 2^(3 * 1/4) = 2^(3/4). This is a classic example of how exponent rules can simplify complex expressions. It’s like using a shortcut to get to your destination faster. By multiplying the exponents, we've reduced the complexity of the denominator and made it easier to compare with the numerator. Think of it as zooming in on a map – by focusing on the relevant details, we can see the path more clearly. This step is particularly important because it allows us to combine the terms with the same base, which is a key strategy in simplifying exponential expressions. It’s like finding a common denominator when adding fractions – it allows us to perform the operation and get to a single, simplified result.

Simplified Expression

After applying these rules, our expression becomes (2^(1/2) * 5^(1/2)) / 2^(3/4). Now we're cooking with gas! We have the same base (2) in both the numerator and the denominator, which means we can use the quotient rule for exponents to simplify further. This is like having two puzzle pieces that fit perfectly together – we're now able to combine them and move closer to the solution. The quotient rule states that when you divide terms with the same base, you subtract the exponents. This is a fundamental rule in algebra, and mastering it will help you tackle a wide range of problems involving exponents and radicals. Think of it as a mathematical Swiss Army knife – it’s a versatile tool that can be used in many different situations. By applying the quotient rule, we're essentially canceling out the common factors and reducing the expression to its simplest form. It’s like pruning a plant – by removing the unnecessary branches, we allow the plant to grow stronger and more efficiently.

Quotient Rule for Exponents

Now for the final simplification! We use the quotient rule for exponents, which says that when dividing terms with the same base, we subtract the exponents. So, 2^(1/2) / 2^(3/4) = 2^(1/2 - 3/4). Let's calculate the exponent: 1/2 - 3/4 = 2/4 - 3/4 = -1/4. This means our expression simplifies to 2^(-1/4) * 5^(1/2). But wait, we're not quite done yet! We need to rewrite this in radical form to match the answer choices. This is like translating from one language to another – we need to express our answer in the correct format. Remember that a negative exponent means we take the reciprocal, so 2^(-1/4) is the same as 1 / 2^(1/4). And 5^(1/2) is just the square root of 5. So, our expression becomes (1 / 2^(1/4)) * √(5). This is getting closer, but we still need to rationalize the denominator. This is a common technique in algebra, and it involves getting rid of the radical in the denominator. It’s like cleaning up your workspace after finishing a project – we want to present our answer in the most polished and professional way possible.

Subtracting Exponents

Applying the quotient rule, we subtract the exponents of the terms with the same base: 2^(1/2) / 2^(3/4) = 2^(1/2 - 3/4). This is a crucial step because it allows us to combine the terms with the same base and simplify the expression. It’s like merging two lanes of traffic into one – we're streamlining the expression and making it easier to manage. Now we need to calculate the exponent: 1/2 - 3/4. To do this, we need a common denominator, which is 4. So, 1/2 becomes 2/4, and the subtraction becomes 2/4 - 3/4 = -1/4. This gives us 2^(-1/4). Remember, a negative exponent means we take the reciprocal, so 2^(-1/4) is the same as 1 / 2^(1/4). This is an important concept to understand because it allows us to move between positive and negative exponents and simplify expressions more effectively. Think of it as having a mathematical superpower – you can flip the expression and change the sign of the exponent. This step is essential for getting our expression into a form that we can easily convert back into radical form.

Resulting Expression

So, after subtracting the exponents, we have 2^(-1/4) * 5^(1/2). This is a significant milestone because we've simplified the expression as much as possible using exponent rules. It’s like reaching the summit of a mountain – we can now see the path clearly and know exactly where we need to go next. However, we're not quite finished yet. We need to rewrite this expression in radical form to match the answer choices. This is like translating a sentence from one language to another – we need to express our answer in the correct format. Remember that a negative exponent means we take the reciprocal, so 2^(-1/4) is the same as 1 / 2^(1/4). And 5^(1/2) is simply the square root of 5. This gives us (1 / 2^(1/4)) * √(5). Now we're getting really close! The final step is to rationalize the denominator, which will give us our final answer.

Rationalizing the Denominator

Our expression is now (1 / 2^(1/4)) * √(5). To rationalize the denominator, we need to get rid of the fourth root of 2 in the denominator. To do this, we'll multiply both the numerator and the denominator by 2^(3/4). Why 2^(3/4)? Because 2^(1/4) * 2^(3/4) = 2^(1/4 + 3/4) = 2^(4/4) = 2^1 = 2, which eliminates the radical in the denominator. This is a classic technique for rationalizing denominators, and it's based on the idea that multiplying a radical by itself (or a suitable power of itself) will eliminate the radical. Think of it as finding the missing piece of a puzzle – we're looking for the term that, when multiplied by the denominator, will give us a whole number. So, we multiply both the numerator and the denominator by 2^(3/4) to maintain the value of the expression. This gives us (√(5) * 2^(3/4)) / (2^(1/4) * 2^(3/4)) = (√(5) * 2^(3/4)) / 2. Now we need to rewrite the numerator in radical form. Remember that 2^(3/4) is the same as the fourth root of 2 cubed, which is the fourth root of 8. So, our expression becomes (√(5) * ∜(8)) / 2. We can rewrite √(5) as ∜(5^2) = ∜(25). Now we have (∜(25) * ∜(8)) / 2. Multiplying the radicals in the numerator, we get ∜(25 * 8) / 2 = ∜(200) / 2. And there's our answer! This process might seem a bit long and complicated, but it's a great example of how we can use exponent rules and radical manipulation to simplify expressions. The key is to break the problem down into smaller steps and tackle each one individually.

Multiplying by the Conjugate

To rationalize the denominator in (1 / 2^(1/4)) * √(5), we need to multiply both the numerator and the denominator by a term that will eliminate the fourth root in the denominator. In this case, we'll multiply by 2^(3/4). Why 2^(3/4)? Because when we multiply 2^(1/4) by 2^(3/4), we get 2^(1/4 + 3/4) = 2^(4/4) = 2^1 = 2, which is a whole number. This is a common technique for rationalizing denominators, and it's based on the idea that multiplying a radical by a suitable term will eliminate the radical. Think of it as finding the perfect partner for a dance – we're looking for the term that, when paired with the denominator, will create a harmonious result. So, we multiply both the numerator and the denominator by 2^(3/4) to maintain the value of the expression. This gives us (√(5) * 2^(3/4)) / (2^(1/4) * 2^(3/4)) = (√(5) * 2^(3/4)) / 2. This step is crucial because it gets rid of the radical in the denominator, which is the goal of rationalization.

Final Simplification

After multiplying by 2^(3/4), our expression becomes (√(5) * 2^(3/4)) / 2. Now we need to rewrite the numerator in radical form. Remember that 2^(3/4) is the same as the fourth root of 2 cubed, which is the fourth root of 8. So, our expression becomes (√(5) * ∜(8)) / 2. To combine the radicals, we need to express them with the same index. We can rewrite √(5) as ∜(5^2) = ∜(25). Now we have (∜(25) * ∜(8)) / 2. Multiplying the radicals in the numerator, we get ∜(25 * 8) / 2 = ∜(200) / 2. And there it is! We've finally simplified the expression and rationalized the denominator. This process might seem a bit long and complicated, but it's a great example of how we can use exponent rules and radical manipulation to solve problems. The key is to break the problem down into smaller steps and tackle each one individually. It’s like climbing a ladder – each step gets us closer to the top.

The Answer

So, after all that simplifying, we find that the expression equivalent to √(10) / ∜(8) is ∜(200) / 2. This matches option A. Whew! That was a journey, but we made it. Remember, guys, the key to these problems is breaking them down into smaller, manageable steps, and using the rules of exponents and radicals to your advantage. Keep practicing, and you'll become a pro at these in no time!

Therefore, the equivalent expression is:

A. ∜(200) / 2

Congratulations, we solved it together! Remember, the journey through each math problem not only sharpens our skills but also enriches our understanding. Keep up the fantastic work, and let's keep exploring the fascinating world of mathematics! Each challenge we tackle is a stepping stone towards greater confidence and mastery. So, keep practicing, keep questioning, and most importantly, keep enjoying the process. Math isn't just about numbers and equations; it's about critical thinking, problem-solving, and the joy of discovery. Let’s continue this journey together, unlocking new mathematical mysteries and expanding our horizons. On to the next challenge!