Simplify √[1]{2x²} ⋅ √[1]{2x³}: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a fascinating problem that involves simplifying expressions with radicals and exponents. Our mission is to find an expression equivalent to √[1]{2x²} ⋅ √[1]{2x³}. This might seem daunting at first, but don't worry, we'll break it down step by step. So, grab your thinking caps, and let's get started!

Understanding the Basics of Radicals and Exponents

Before we jump into the problem, let's refresh our understanding of radicals and exponents. These are fundamental concepts in algebra, and mastering them is crucial for solving more complex problems. Think of radicals and exponents as two sides of the same coin – they're closely related and often used together.

What are Radicals?

Radicals, often represented by the symbol '√', are used to denote roots of numbers. The most common type is the square root (√), which asks: "What number, when multiplied by itself, equals the number under the radical?" For example, √9 = 3 because 3 * 3 = 9. But radicals aren't limited to square roots. We can also have cube roots (∛), fourth roots (∜), and so on. The small number in the crook of the radical symbol (like the '3' in ∛) is called the index, and it tells us which root we're looking for. In our problem, we have expressions with a radical, so understanding how they work is key.

The Power of Exponents

Exponents, on the other hand, provide a concise way to express repeated multiplication. For example, x³ means x * x * x. The base (x in this case) is the number being multiplied, and the exponent (3) tells us how many times to multiply it. Exponents can also be fractions, which is where the connection to radicals becomes clear. A fractional exponent represents both a power and a root. For instance, x^(1/2) is the same as √x, and x^(1/3) is the same as ∛x. This relationship is vital for simplifying expressions like the one we're tackling today.

Radicals and Fractional Exponents: A Perfect Match

The relationship between radicals and fractional exponents is the key to simplifying many algebraic expressions. The general rule is: √[n]{x^m} = x^(m/n). In other words, the nth root of x raised to the power of m is the same as x raised to the power of m/n. This rule allows us to convert between radical form and exponential form, making it easier to manipulate expressions.

For example, let's say we have √[3]{x^6}. Using the rule, we can rewrite this as x^(6/3), which simplifies to x². This conversion is incredibly useful when dealing with multiplication and division of radicals, as we'll see in our problem.

Understanding these basics is like having the right tools for the job. Now that we've got our tools sharpened, let's dive into the main problem and see how we can use these concepts to find the equivalent expression.

Breaking Down the Problem: √[1]{2x²} ⋅ √[1]{2x³}

Okay, guys, let's tackle the expression √[1]{2x²} ⋅ √[1]{2x³}. At first glance, it might seem a bit intimidating, but don't worry, we're going to break it down into manageable steps. The key here is to use the properties of radicals and exponents that we just discussed. Remember, the goal is to simplify the expression and find an equivalent form among the given options.

Step 1: Rewrite Radicals as Fractional Exponents

The first step in simplifying this expression is to rewrite the radicals as fractional exponents. This makes it easier to combine the terms and apply the rules of exponents. Recall that √[n]{x^m} is equivalent to x^(m/n). In our case, we have √[1]{2x²} and √[1]{2x³}.

Let's start with √[1]{2x²}. This can be rewritten as (2x²)^(1/1), which is simply 2x². Similarly, √[1]{2x³} can be rewritten as (2x³)^(1/1), which is 2x³. Now our expression looks like this: 2x² ⋅ 2x³.

Step 2: Combine Like Terms

Now that we've converted the radicals to exponential form, we can combine the terms. Remember the rule for multiplying exponents with the same base: x^a ⋅ x^b = x^(a+b). This means we can add the exponents of the 'x' terms together.

So, we have 2x² ⋅ 2x³. First, multiply the coefficients (the numbers in front of the 'x' terms): 2 * 2 = 4. Next, multiply the 'x' terms: x² ⋅ x³ = x^(2+3) = x^5. Combining these, we get 4x^5.

Step 3: Convert Back to Radical Form (if necessary)

In some cases, you might need to convert the expression back to radical form. However, in this case, our simplified expression 4x^5 is already in a pretty straightforward form. We've successfully combined the terms and simplified the original expression.

Step 4: Compare with the Given Options

Now that we've simplified the expression to 4x^5, we need to compare it with the given options to find the equivalent one. The options are:

A. 2^(1/2) x^(6/2) B. 2^(2/4) x^(5/4) C. 4^(1/3) x^(5/2)

Let's analyze each option to see which one matches our simplified expression.

Analyzing the Options: Finding the Equivalent Expression

Alright, let's put on our detective hats and analyze each option to see which one is equivalent to our simplified expression, 4x^5. This step is crucial because it tests our ability to manipulate exponents and recognize equivalent forms. Remember, the correct answer might not look exactly like 4x^5 at first glance, so we need to be ready to do some further simplification.

Option A: 2^(1/2) x^(6/2)

Option A is 2^(1/2) x^(6/2). Let's simplify this piece by piece. First, 2^(1/2) is the same as √2, which is an irrational number approximately equal to 1.414. Next, x^(6/2) simplifies to x^3 because 6/2 = 3. So, Option A is √2 * x^3. This doesn't match our simplified expression of 4x^5, so Option A is not the correct answer.

Option B: 2^(2/4) x^(5/4)

Option B is 2^(2/4) x^(5/4). Again, let's simplify. 2^(2/4) can be simplified to 2^(1/2), which, as we know, is √2. The term x^(5/4) is a bit trickier. We can rewrite it as x^(1 + 1/4) or x^1 * x^(1/4), which is x * ∜x. So, Option B is √2 * x * ∜x. This also doesn't match our simplified expression of 4x^5, making Option B incorrect.

Option C: 4^(1/2) x^(5/2)

Wait a minute, guys! There seems to be a typo in the options provided. None of them simplify to 4x^5. It looks like we've hit a snag in our quest to find the equivalent expression. But don't worry, this is a common occurrence in math problems, and it gives us an opportunity to think critically about the process we've followed.

Addressing the Discrepancy

Since none of the options match our simplified expression, it's important to double-check our work and the original problem. Sometimes, a small error in the initial steps can lead to a different final answer. We've carefully gone through each step, but it's always good to have a fresh look.

Conclusion: The Importance of Careful Analysis

In conclusion, while we couldn't find a matching option due to a potential error in the choices provided, we successfully simplified the original expression √[1]{2x²} ⋅ √[1]{2x³} to 4x^5. This exercise highlights the importance of understanding the fundamentals of radicals and exponents, as well as the need for careful analysis and step-by-step simplification. Remember, math isn't just about getting the right answer; it's about the process of problem-solving and critical thinking. Keep practicing, and you'll become a master of mathematical expressions!