What Is The Simplified Form Of The Expression (4x√5x² + 2x²√6)² Given That X ≥ 0?
In the realm of mathematics, simplifying complex expressions is a fundamental skill. Today, we embark on a journey to dissect and simplify the product (4x√5x² + 2x²√6)², where x ≥ 0. This exploration will not only reveal the final simplified form but also illuminate the underlying principles of algebraic manipulation and radical simplification. Let's dive in!
Understanding the Expression
The expression (4x√5x² + 2x²√6)² appears intricate at first glance. It involves a binomial (a two-term expression) raised to the power of 2. To unravel this, we'll employ the well-known algebraic identity: (a + b)² = a² + 2ab + b². This identity provides a roadmap for expanding the given expression into a more manageable form.
Before we jump into the expansion, let's identify the 'a' and 'b' components in our expression. Here, a = 4x√5x² and b = 2x²√6. Now, we have a clear framework for applying the (a + b)² identity. The expansion process involves squaring each term individually (a² and b²) and then adding twice the product of the two terms (2ab). This methodical approach ensures that we capture all components of the expanded expression.
Breaking Down the Terms
To effectively apply the (a + b)² identity, we need to carefully consider each term. The term 4x√5x² involves a product of a constant (4), a variable (x), and a radical expression (√5x²). The radical √5x² can be further simplified by recognizing that x² is a perfect square. This allows us to extract x from the radical, leading to √5 * √(x²) = √5 * |x|. Since we are given that x ≥ 0, we can simplify |x| to x. Therefore, 4x√5x² becomes 4x * x√5 = 4x²√5.
Similarly, the term 2x²√6 involves a constant (2), a variable term (x²), and a radical (√6). Since 6 does not have any perfect square factors (other than 1), the radical √6 remains in its simplest form. Consequently, the term 2x²√6 is already in a simplified state, ready to be used in our expansion.
By simplifying each term individually, we have made the subsequent expansion process significantly easier. We have effectively broken down the complex expression into manageable components, paving the way for a smooth application of the (a + b)² identity.
Expanding the Expression Using (a + b)²
Now that we have identified and simplified our 'a' and 'b' terms (a = 4x²√5 and b = 2x²√6), we can proceed with the expansion using the (a + b)² = a² + 2ab + b² identity. This is the heart of our simplification process, where we transform the original expression into a sum of individual terms.
First, let's calculate a². This means squaring the term 4x²√5. Squaring each factor, we get (4x²√5)² = 4² * (x²)² * (√5)² = 16 * x⁴ * 5 = 80x⁴. This represents the first component of our expanded expression.
Next, we calculate b². This involves squaring the term 2x²√6. Following the same procedure, we get (2x²√6)² = 2² * (x²)² * (√6)² = 4 * x⁴ * 6 = 24x⁴. This is the second component of our expanded expression.
Finally, we calculate 2ab. This means multiplying 2 by the product of a and b. So, 2ab = 2 * (4x²√5) * (2x²√6) = 2 * 4 * 2 * x² * x² * √5 * √6 = 16x⁴√(5 * 6) = 16x⁴√30. This represents the third and final component of our expanded expression.
By meticulously calculating each component (a², b², and 2ab), we have successfully expanded the original expression. We have transformed the binomial squared into a trinomial (a three-term expression), which is now ready for further simplification by combining like terms.
Simplifying the Expanded Expression
With the expression expanded as 80x⁴ + 24x⁴ + 16x⁴√30, our next step is to simplify it by combining like terms. Like terms are those that have the same variable and exponent. In our expanded expression, we have two terms with x⁴ (80x⁴ and 24x⁴) and one term with x⁴√30 (16x⁴√30). Combining like terms is a fundamental algebraic technique that allows us to reduce the number of terms in an expression and present it in a more concise form.
To combine the x⁴ terms, we simply add their coefficients: 80x⁴ + 24x⁴ = (80 + 24)x⁴ = 104x⁴. This combines the first two terms into a single term. The third term, 16x⁴√30, cannot be combined with the other terms because it contains the radical √30. Terms with radicals can only be combined if they have the same radical component.
Therefore, the simplified expression is 104x⁴ + 16x⁴√30. This is the final simplified form of the original expression (4x√5x² + 2x²√6)². We have successfully navigated the expansion and simplification process, arriving at a clear and concise result.
Key Takeaways
Throughout this exploration, we have employed several key mathematical principles and techniques. We utilized the (a + b)² identity to expand the squared binomial, simplified radical expressions by extracting perfect square factors, and combined like terms to arrive at the final simplified form. These techniques are fundamental to algebraic manipulation and are applicable to a wide range of mathematical problems.
Moreover, we emphasized the importance of methodical steps in simplifying complex expressions. By breaking down the problem into smaller, manageable steps, we minimized the chances of errors and ensured a clear path to the solution. This systematic approach is a valuable asset in any mathematical endeavor.
Conclusion: The Simplified Product
In conclusion, the simplified form of the product (4x√5x² + 2x²√6)², where x ≥ 0, is 104x⁴ + 16x⁴√30. This final expression represents the culmination of our algebraic journey, showcasing the power of expansion, simplification, and methodical problem-solving. This journey reinforces the importance of understanding fundamental algebraic identities and techniques in simplifying complex mathematical expressions. The ability to manipulate and simplify expressions is a cornerstone of mathematical proficiency, empowering us to tackle more intricate problems with confidence and precision.
What is the simplified form of the expression (4x√5x² + 2x²√6)² given that x ≥ 0?
Simplify (4x√5x² + 2x²√6)² A Step-by-Step Guide with Solution