Equivalent Expression Of (2^(1/2) * 2^(3/4))^2

Hey guys! Let's break down this math problem together. We're going to figure out which expression is the same as $\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2$. It looks a bit intimidating at first, but don't worry, we'll take it step by step. We'll go through the properties of exponents and radicals to simplify the expression and match it with the correct option. This kind of problem is super common in algebra, so understanding how to solve it will definitely help you out in the future. Let's get started and make math a little less scary and a lot more fun!

Understanding the Problem

Before we dive into solving, let's make sure we understand what the question is asking. The core of the problem lies in simplifying the given expression, which involves exponents and a power of a product. We need to use the rules of exponents to combine the terms inside the parentheses first, and then apply the outer exponent. The question presents us with multiple choices in radical form, meaning we'll also need to know how to convert between exponential and radical forms. This involves understanding the relationship between fractional exponents and roots. Remember, $x^{\frac{a}{b}}$ is the same as $\sqrt[b]{x^a}$. Keeping this in mind will be crucial as we simplify and compare our result with the given options. So, to recap, our mission is to simplify the expression and express it in a way that matches one of the provided radical forms. This requires a solid grasp of exponent rules and the connection between exponents and radicals. Now that we've clarified the goal, let's roll up our sleeves and start simplifying!

Step-by-Step Solution

Alright, let's get into the nitty-gritty and solve this step-by-step. This is where we'll put those exponent rules into action! Remember, the key is to take it one step at a time and keep track of what we're doing.

1. Combine the terms inside the parentheses: We have $2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}$. When multiplying terms with the same base, we add the exponents. So, we need to add $\frac{1}{2}$ and $\frac{3}{4}$.

   • To add these fractions, we need a common denominator, which is 4. So, we rewrite $\frac{1}{2}$ as $\frac{2}{4}$.
   • Now we have $\frac{2}{4} + \frac{3}{4} = \frac{5}{4}$.
   • Therefore, $2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}} = 2^{\frac{5}{4}}$.

2. Apply the outer exponent: Now we have $\left(2^{\frac{5}{4}}\right)^2$. When raising a power to another power, we multiply the exponents.

   • So, we multiply $\frac{5}{4}$ by 2, which gives us $\frac{5}{4} \cdot 2 = \frac{10}{4}$.
   • We can simplify $\frac{10}{4}$ to $\frac{5}{2}$.
   • Thus, $\left(2^{\frac{5}{4}}\right)^2 = 2^{\frac{5}{2}}$.

3. Convert to radical form: Now we need to convert $2^{\frac{5}{2}}$ to radical form. Remember that $x^{\frac{a}{b}}$ is the same as $\sqrt[b]{x^a}$.

   • In our case, we have $2^{\frac{5}{2}}$, so $a = 5$ and $b = 2$.
   • This means $2^{\frac{5}{2}} = \sqrt[2]{2^5}$, which is simply $\sqrt{2^5}$ (since the index of 2 is usually omitted for square roots).

So, after all those steps, we've found that $\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2$ is equivalent to $\sqrt{2^5}$. Now, let’s see which of the answer choices matches our simplified expression.

Matching the Solution with the Options

Okay, we've simplified the expression to $\sqrt{2^5}$. Now the fun part begins – matching our answer with the given options! This is like a mini-puzzle within the bigger problem. We need to carefully look at each option and see if it's just another way of writing $\sqrt{2^5}$. Sometimes, the options might be simplified in a slightly different way, so we might need to do a little more manipulation to see the match. This step is crucial because it ensures we're not just stopping at the right simplification, but also recognizing it in different forms. So, let's dive into those options and see which one clicks!

• A. $\sqrt[4]{2^3}$: This doesn't look like our answer at all. The index of the radical is different, and the exponent inside is different. So, we can rule this one out pretty quickly.
• B. $\sqrt{2^5}$: Ding ding ding! This is exactly what we got! $\sqrt{2^5}$ is our simplified answer. We've found a match!
• C. $\sqrt[4]{4^3}$: This one looks a bit trickier. We could try rewriting $4$ as $2^2$ to see if it gets us closer to our answer. So, $\sqrt[4]{4^3} = \sqrt[4]{(2^2)^3} = \sqrt[4]{2^6}$. This is not the same as $\sqrt{2^5}$, so we can eliminate this option.
• D. $\sqrt{4^5}$: Again, let's rewrite $4$ as $2^2$. So, $\sqrt{4^5} = \sqrt{(2^2)^5} = \sqrt{2^{10}}$. This is definitely not the same as $\sqrt{2^5}$, so we can rule this out too.

It looks like option B is our winner! We've successfully simplified the expression and found the matching option. High five!

Final Answer

After simplifying the expression $\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2$ and comparing it with the given options, we've confidently arrived at the answer.

The equivalent expression is B. $\sqrt{2^5}$.

So, there you have it! We've tackled this problem step by step, using the properties of exponents and radicals. It's all about breaking down the problem into smaller, manageable steps and applying the rules you know. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time. Great job, guys! You nailed it!