Simplify Exponents And Roots \(2^(1/2) * 2^(3/4))^2 \)

Hey there, math enthusiasts! Today, we're going to unravel a fascinating expression: ${\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2 }$. This might look a bit intimidating at first glance, but don't worry, we'll break it down step by step. Our mission is to figure out which of the given options – A. ${\sqrt[4]{2^3}}$, B. ${\sqrt{2^5}}$, C. ${\sqrt[4]{4^3}}$, and D. ${\sqrt{4^5}}$ – is equivalent to the original expression. So, grab your thinking caps, and let's dive in!

Understanding the Basics: Exponents and Roots

Before we tackle the main expression, let's quickly brush up on some fundamental concepts. We're dealing with exponents and roots here, so it's crucial to have a solid understanding of what they mean and how they interact with each other. At its heart, exponents represent repeated multiplication. For example, ${2^3}$ means 2 multiplied by itself three times (2 * 2 * 2 = 8). The base (in this case, 2) is the number being multiplied, and the exponent (in this case, 3) indicates how many times the base is multiplied by itself. Now, fractional exponents, like those in our expression, represent roots. A root is the inverse operation of an exponent. For instance, the square root of a number (denoted by ${\sqrt{\ }}$) is a value that, when multiplied by itself, equals the original number. Mathematically, ${\sqrt{x} = x^{\frac{1}{2}}}$. Similarly, the cube root (denoted by ${\sqrt[3]{\ }}$) is a value that, when multiplied by itself three times, equals the original number, and so on. The general form is ${\sqrt[n]{x} = x^{\frac{1}{n}}}$, where 'n' is the index of the root. When we see fractional exponents like ${\frac{1}{2}}$ or ${\frac{3}{4}}$, we can interpret them as roots. ${2^{\frac{1}{2}}}$ is the square root of 2, and ${2^{\frac{3}{4}}}$ can be thought of as the fourth root of 2 cubed. Understanding these basics is essential for simplifying and manipulating expressions like the one we're working with.

Step-by-Step Simplification: Unraveling the Expression

Okay, let's get our hands dirty and start simplifying the expression ${\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2 }$. The first thing we need to do is tackle the expression inside the parentheses. We have ${2^{\frac{1}{2}}}$ multiplied by ${2^{\frac{3}{4}}}$. Remember the rule of exponents that says when you multiply numbers with the same base, you add the exponents: ${a^m \cdot a^n = a^{m+n}}$. Applying this rule, we get:

${2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}} = 2^{\frac{1}{2} + \frac{3}{4}}}$

Now, we need to add the fractions ${\frac{1}{2}}$ and ${\frac{3}{4}}$. To do this, we need a common denominator, which in this case is 4. So, we rewrite ${\frac{1}{2}}$ as ${\frac{2}{4}}$. Now we can add:

${\frac{2}{4} + \frac{3}{4} = \frac{5}{4}}$

So, our expression inside the parentheses simplifies to:

${2^{\frac{5}{4}}}$

Now, we have to deal with the exponent outside the parentheses, which is 2. We're raising ${2^{\frac{5}{4}}}$ to the power of 2. Another rule of exponents comes into play here: when you raise a power to another power, you multiply the exponents: ${(a^m)^n = a^{m \cdot n}}$. Applying this rule, we get:

${\left(2^{\frac{5}{4}}\right)^2 = 2^{\frac{5}{4} \cdot 2}}$

Multiplying ${\frac{5}{4}}$ by 2, we get:

${\frac{5}{4} \cdot 2 = \frac{10}{4}}$

We can simplify ${\frac{10}{4}}$ by dividing both the numerator and denominator by 2, which gives us ${\frac{5}{2}}$. So, our expression simplifies to:

${2^{\frac{5}{2}}}$

We've successfully simplified the original expression to ${2^{\frac{5}{2}}}$. This is a crucial step, and it shows how applying the rules of exponents can help us break down complex expressions into manageable forms. Make sure you're comfortable with each of these steps, as they're fundamental to solving this type of problem.

Matching the Simplified Form: Connecting Exponents and Roots

Great job, guys! We've simplified the original expression ${\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2 }$ to ${2^{\frac{5}{2}}}$. Now, the next step is to figure out which of the given options (A, B, C, and D) matches this simplified form. Remember, the options are presented in radical form (with square roots and fourth roots), so we need to bridge the gap between our exponential form and the radical notation. Let's revisit how fractional exponents and roots are related. We know that ${x^{\frac{1}{n}} = \sqrt[n]{x}}$. This means that a fractional exponent of ${\frac{1}{2}}$ corresponds to a square root, ${\frac{1}{3}}$ corresponds to a cube root, ${\frac{1}{4}}$ corresponds to a fourth root, and so on. Our simplified expression is ${2^{\frac{5}{2}}}$. We can rewrite this as ${2^{\frac{5}{2}} = (2^5)^{\frac{1}{2}}}$. Do you see where we're going with this? We've separated the exponent into a whole number (5) and a fraction (${\frac{1}{2}}$). Now, we can apply our knowledge of fractional exponents and roots. ${(2^5)^{\frac{1}{2}}}$ is the same as taking the square root of ${2^5}$. So, we have:

${2^{\frac{5}{2}} = \sqrt{2^5}}$

This is a key transformation, as it directly connects our simplified exponential form to a radical form. We've successfully expressed ${2^{\frac{5}{2}}}$ as ${\sqrt{2^5}}$. Now, let's quickly look at our options again. Option B is ${\sqrt{2^5}}$. Bingo! We have a match. This means that the expression ${\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2 }$ is equivalent to ${\sqrt{2^5}}$. But hold on, we're not done yet. It's always a good idea to double-check and make sure the other options are indeed incorrect. This will solidify our understanding and prevent any silly mistakes. We'll take a quick look at options A, C, and D to confirm that they are not equivalent to ${2^{\frac{5}{2}}}$.

Eliminating Incorrect Options: A Thorough Check

Alright, let's make sure we've got this nailed down. We've identified that option B, ${\sqrt{2^5}}$, is equivalent to our simplified expression ${2^{\frac{5}{2}}}$. Now, we're going to take a quick look at the other options – A, C, and D – to confirm that they are not equivalent. This is a crucial step in problem-solving, as it helps us avoid errors and strengthens our understanding. Let's start with option A: ${\sqrt[4]{2^3}}$. We can rewrite this in exponential form as ${2^{\frac{3}{4}}}$. This is clearly different from our simplified expression, ${2^{\frac{5}{2}}}$, so option A is incorrect. Next up is option C: ${\sqrt[4]{4^3}}$. This looks a bit trickier, but we can handle it. First, let's rewrite 4 as ${2^2}$. So, we have ${\sqrt[4]{(2^2)^3}}$. Using the power of a power rule, we get ${\sqrt[4]{2^6}}$. Now, we can rewrite this in exponential form as ${2^{\frac{6}{4}}}$, which simplifies to ${2^{\frac{3}{2}}}$. This is also different from ${2^{\frac{5}{2}}}$, so option C is incorrect. Finally, let's look at option D: ${\sqrt{4^5}}$. Again, we rewrite 4 as ${2^2}$, giving us ${\sqrt{(2^2)^5}}$. Using the power of a power rule, we get ${\sqrt{2^{10}}}$. Now, we rewrite this in exponential form as ${2^{\frac{10}{2}}}$, which simplifies to ${2^5}$. This is definitely not the same as ${2^{\frac{5}{2}}}$, so option D is incorrect. Phew! We've thoroughly checked all the options and confirmed that only option B, ${\sqrt{2^5}}$, is equivalent to the original expression. This comprehensive approach ensures that we're not just guessing, but truly understanding the problem and its solution. It's a great habit to develop for tackling any math problem.

Concluding Thoughts: Mastering Exponents and Roots

Woo-hoo! We did it, guys! We successfully decoded the expression ${\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2 }$ and determined that it is equivalent to ${\sqrt{2^5}}$. We walked through the step-by-step simplification process, applying the rules of exponents and understanding the relationship between fractional exponents and roots. We also took the extra step of eliminating the incorrect options to solidify our understanding. This journey highlights the importance of mastering the fundamentals of exponents and roots. These concepts are not just isolated topics; they are building blocks for more advanced mathematical concepts. By understanding how exponents work, how to manipulate them, and how they relate to roots, you'll be well-equipped to tackle a wide range of mathematical problems. Remember, practice makes perfect! The more you work with exponents and roots, the more comfortable you'll become with them. Try solving similar problems, experimenting with different expressions, and challenging yourself to think critically. Math isn't just about memorizing formulas; it's about understanding the underlying principles and applying them creatively. So, keep exploring, keep learning, and keep having fun with math! You've got this!