Simplify (4x√(5x²) + 2x²√6)²: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving deep into an interesting algebraic expression: (4x√(5x²) + 2x²√6)², with the condition that x ≥ 0. At first glance, it might seem a bit intimidating, but don't worry, we'll break it down step by step. Our goal is to simplify this expression and understand its components. We will explore the intricacies of radicals, exponents, and algebraic manipulations. By the end of this journey, you'll not only know what this expression represents but also how to tackle similar problems with confidence. So, let's put on our thinking caps and get started!

Understanding the Components

Before we jump into simplifying the entire expression, let's dissect it into smaller, more manageable parts. This will help us understand the role each component plays and how they interact with each other. First, we have the variable x, which represents a non-negative number (x ≥ 0). This condition is crucial because it affects how we handle square roots and ensures we're dealing with real numbers. Inside the parentheses, we have two terms: 4x√(5x²) and 2x²√6. The first term involves a square root containing x², while the second term has a constant under the square root. Remember, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Also, let's not forget the exponent of 2 outside the parentheses, which means we're squaring the entire expression inside. This means we will be multiplying the entire expression by itself. Understanding these basic components is crucial for simplifying the expression effectively. We'll start by simplifying the square root term and then move on to combining the terms and squaring the result. So, stay with me, guys, as we unravel this mathematical puzzle!

Simplifying the Square Root

The most immediate simplification we can make is within the square root. Notice the term √(5x²). We can rewrite this using the properties of square roots. Recall that √(ab) = √a * √b. Applying this property, we can separate the square root as follows: √(5x²) = √5 * √x². Now, here's where the condition x ≥ 0 becomes important. Since x is non-negative, the square root of x² is simply x. This is because the square root function always returns the non-negative root. So, √x² = x (when x ≥ 0). If we didn't have this condition, we would need to consider the absolute value, |x|, which adds a layer of complexity. But thankfully, with x ≥ 0, we can directly replace √x² with x. Therefore, our simplified term becomes √5 * x, or x√5. This simplification is a key step because it removes the square from under the square root, making the expression much easier to work with. Now, let's substitute this back into our original expression and see how it looks. The expression now becomes (4x * x√5 + 2x²√6)², which is already looking less intimidating, right? We've successfully tackled the square root part, and now we're ready to move on to the next phase of simplification. Keep your eyes peeled, because the magic of algebra is about to unfold even further!

Combining Like Terms

Now that we've simplified the square root, let's focus on the terms inside the parentheses. Our expression currently looks like this: (4x * x√5 + 2x²√6)². The first thing we can do is simplify the first term by multiplying the x terms: 4x * x√5 = 4x²√5. So, now our expression looks like (4x²√5 + 2x²√6)². Notice anything interesting? Both terms inside the parentheses now have a common factor of x². This is a fantastic opportunity to factor out x² and simplify the expression even further. Factoring out x² gives us: (x²(4√5 + 2√6))². Factoring is a powerful technique in algebra because it allows us to rewrite expressions in a more manageable form. In this case, by factoring out x², we've separated the variable part from the constant part. This makes it easier to see the structure of the expression and to apply the exponent. Now, we have a product inside the parentheses, and we're squaring the entire product. Remember the rule of exponents: (ab)² = a²b². We can apply this rule to our expression, which means we'll square both x² and the constant term (4√5 + 2√6). This will lead us to the next step, where we'll handle the squaring of the constant term. So, let's keep pushing forward; we're making great progress in unraveling this expression!

Squaring the Expression

Following the rule (ab)² = a²b², we can rewrite our expression (x²(4√5 + 2√6))² as (x²)² * (4√5 + 2√6)². Squaring x² is straightforward: (x²)² = x^(22) = x⁴. So, we have x⁴ multiplied by the square of the binomial (4√5 + 2√6). Now, the real work begins: we need to square (4√5 + 2√6). To do this, we'll use the formula (a + b)² = a² + 2ab + b². In our case, a = 4√5 and b = 2√6. Let's calculate each term: a² = (4√5)² = 4² * (√5)² = 16 * 5 = 80 b² = (2√6)² = 2² * (√6)² = 4 * 6 = 24 2ab = 2 * (4√5) * (2√6) = 16√5√6 = 16√(56) = 16√30 Now, let's put it all together: (4√5 + 2√6)² = 80 + 16√30 + 24 = 104 + 16√30. Remember, we can only add like terms, so we combine the constants 80 and 24. We cannot combine the constant term 104 with the term 16√30 because it involves a square root. Finally, we multiply this result by x⁴ to get our fully simplified expression: x⁴(104 + 16√30). Guys, we've made it! We've successfully squared the binomial and combined it with the x⁴ term. This expression is now in its simplest form, and we've tackled a pretty complex algebraic problem. Pat yourselves on the back for following along!

Final Simplified Form

After all the algebraic maneuvering, we've arrived at the final simplified form of our expression: x⁴(104 + 16√30). This is a much cleaner and more understandable representation of the original expression, (4x√(5x²) + 2x²√6)². By breaking down the problem into smaller steps, we were able to tackle the square root, combine like terms, and square the binomial. We used key algebraic principles like the distributive property, factoring, and the rules of exponents. This process highlights the importance of understanding the individual components of an expression and how they interact with each other. Remember, in mathematics, complexity often hides underlying simplicity. The key is to systematically unpack the layers, apply the appropriate rules, and simplify along the way. Now, let's take a moment to appreciate what we've accomplished. We started with a seemingly complicated expression and, through careful analysis and step-by-step simplification, we transformed it into a more elegant and manageable form. This is the power of algebra – to reveal the hidden structure and beauty within mathematical expressions. So, next time you encounter a daunting equation, remember our journey here, and approach it with confidence and a systematic mindset!

Conclusion

In conclusion, we successfully simplified the expression (4x√(5x²) + 2x²√6)² where x ≥ 0 to its final form: x⁴(104 + 16√30). We started by understanding the individual components of the expression, then simplified the square root, combined like terms, and finally, squared the binomial. This process not only gave us the simplified form but also reinforced our understanding of key algebraic principles. We saw how the condition x ≥ 0 played a crucial role in simplifying the square root term. We also utilized techniques like factoring and the rules of exponents to make the expression more manageable. Guys, remember that simplifying expressions like this is not just about getting the right answer; it's about developing problem-solving skills and a deeper understanding of mathematical concepts. Each step we took, from simplifying the square root to squaring the binomial, contributed to our overall understanding. So, keep practicing, keep exploring, and keep challenging yourselves with new mathematical problems. The more you practice, the more comfortable and confident you'll become in your ability to tackle complex expressions. And who knows, maybe you'll even start to enjoy the process of unraveling these mathematical puzzles! Until next time, keep those mathematical gears turning!