Simplifying Algebraic Expressions A Step-by-Step Guide

Navigating the realm of algebraic expressions often requires us to simplify them into their most basic form. This not only makes them easier to understand but also facilitates further calculations and manipulations. In this article, we will delve into simplifying a specific algebraic expression involving square roots and variables. Our focus will be on the expression ${\frac{\sqrt{12 x^3}}{\sqrt{3 x^2}}}$ where ${x \geq 0}$. We will explore the step-by-step process of simplification, highlighting the key concepts and rules of exponents and radicals that come into play. By the end of this discussion, you'll have a clear understanding of how to tackle similar problems and confidently arrive at the simplest form of such expressions. Let's embark on this mathematical journey together!

Understanding the Problem

Before we dive into the solution, it's crucial to understand the problem at hand. We are presented with the expression ${\frac{\sqrt{12 x^3}}{\sqrt{3 x^2}}}$ and tasked with simplifying it. The condition ${x \geq 0}$ is important because it ensures that we are dealing with non-negative values under the square root, which is a necessary condition for real number solutions. This expression involves square roots, variables raised to powers, and a fraction, so we'll need to employ several algebraic techniques to simplify it effectively. The goal is to transform the given expression into a form that is both mathematically equivalent and easier to work with. This often means reducing the coefficients, simplifying the variables' exponents, and eliminating any unnecessary radicals. Let's break down the expression and identify the different components we need to address.

Breaking Down the Expression

To begin, let's break down the expression into its constituent parts. We have a fraction where both the numerator and the denominator contain square roots. The numerator is ${\sqrt{12 x^3}}$, and the denominator is ${\sqrt{3 x^2}}$. Each of these parts can be further simplified by considering the properties of square roots and exponents. For the numerator, we can recognize that 12 can be factored into ${4 \times 3}$, and 4 is a perfect square. Similarly, ${x^3}$ can be written as ${x^2 \times x}$, where ${x^2}$ is a perfect square. In the denominator, we have ${\sqrt{3 x^2}}$, where ${x^2}$ is also a perfect square. These observations provide us with a roadmap for simplifying the expression. We can use the property ${\sqrt{ab} = \sqrt{a} \times \sqrt{b}}$ to separate the perfect squares from the remaining terms and then simplify the square roots of the perfect squares. This will lead us closer to the simplest form of the expression.

Step-by-Step Simplification

Now, let's proceed with the step-by-step simplification of the expression. We'll start by applying the properties of square roots to both the numerator and the denominator.

1. Simplify the numerator:

   • We have ${\sqrt{12 x^3}}$. As we noted earlier, 12 can be factored into ${4 \times 3}$, and ${x^3}$ can be written as ${x^2 \times x}$. So, we can rewrite the numerator as ${\sqrt{4 \times 3 \times x^2 \times x}}$.
   • Using the property ${\sqrt{ab} = \sqrt{a} \times \sqrt{b}}$, we can separate the square root: ${\sqrt{4} \times \sqrt{3} \times \sqrt{x^2} \times \sqrt{x}}$.
   • Now, we simplify the square roots of the perfect squares: ${2 \times \sqrt{3} \times x \times \sqrt{x}}$. This gives us ${2x \sqrt{3x}}$.

2. Simplify the denominator:

   • We have ${\sqrt{3 x^2}}$. Using the property ${\sqrt{ab} = \sqrt{a} \times \sqrt{b}}$, we can separate the square root: ${\sqrt{3} \times \sqrt{x^2}}$.
   • Simplifying the square root of the perfect square, we get ${\sqrt{3} \times x}$.

3. Rewrite the expression:

   • Now that we've simplified the numerator and the denominator, we can rewrite the original expression as ${\frac{2x \sqrt{3x}}{x \sqrt{3}}}$ . This form is much simpler than the original, but we can still do more to reduce it.

Combining and Reducing

With the numerator and denominator simplified, the next step is to combine and reduce the fraction. We now have the expression ${\frac{2x \sqrt{3x}}{x \sqrt{3}}}$ . Notice that we have common factors in both the numerator and the denominator that can be canceled out. This process of canceling common factors is a fundamental technique in simplifying fractions, and it allows us to arrive at the most concise form of the expression.

1. Cancel the common factors:
   • We can see that ${x}$ is a common factor in both the numerator and the denominator. So, we can cancel out ${x}$ from both parts. Also, notice that ${\sqrt{3}}$ appears in both the numerator (within ${\sqrt{3x}}$) and the denominator. We will address that in the next substep. After canceling out the ${x}$, we are left with ${\frac{2 \sqrt{3x}}{\sqrt{3}}}$ .
2. Further simplification of radicals:
   • We can rewrite ${\sqrt{3x}}$ as ${\sqrt{3} \times \sqrt{x}}$. Now, our expression looks like ${\frac{2 \sqrt{3} \sqrt{x}}{\sqrt{3}}}$ .
   • We now have ${\sqrt{3}}$ as a common factor in both the numerator and the denominator, so we can cancel it out. This leaves us with ${2 \sqrt{x}}$.

By systematically simplifying the numerator and denominator and then canceling common factors, we have reduced the original expression to a much simpler form. This process highlights the importance of recognizing perfect squares and applying the properties of square roots to efficiently simplify algebraic expressions.

Final Simplified Form

After the step-by-step simplification process, we've arrived at the final simplified form of the expression. The original expression, ${\frac{\sqrt{12 x^3}}{\sqrt{3 x^2}}}$ , has been reduced to ${2 \sqrt{x}}$. This result is significantly cleaner and easier to work with than the initial expression. It demonstrates the power of algebraic simplification in making complex expressions more manageable. The final simplified form not only provides a more concise representation of the original expression but also makes it easier to evaluate or use in further calculations. It is essential to present the expression in its simplest form to avoid ambiguity and ensure clear communication of mathematical ideas.

Checking the Answer

To ensure that our simplification is correct, it's always a good practice to check the answer. One way to do this is to substitute a value for ${x}$ into both the original expression and the simplified expression and see if we get the same result. Let's choose a simple value, say ${x = 4}$, since it's a perfect square and will simplify nicely under the square root.

1. Original expression:

   • ${\frac{\sqrt{12 x^3}}{\sqrt{3 x^2}} = \frac{\sqrt{12 (4)^3}}{\sqrt{3 (4)^2}} = \frac{\sqrt{12 \times 64}}{\sqrt{3 \times 16}} = \frac{\sqrt{768}}{\sqrt{48}}}$ .
   • We can simplify ${\sqrt{768}}$ as ${\sqrt{256 \times 3} = 16 \sqrt{3}}$ and ${\sqrt{48}}$ as ${\sqrt{16 \times 3} = 4 \sqrt{3}}$. So, the original expression becomes ${\frac{16 \sqrt{3}}{4 \sqrt{3}} = 4}$.

2. Simplified expression:

   • ${2 \sqrt{x} = 2 \sqrt{4} = 2 \times 2 = 4}$.

Since both expressions evaluate to the same value when ${x = 4}$, this gives us confidence that our simplification is correct. While this check doesn't guarantee correctness for all values of ${x}$, it provides strong evidence that we've simplified the expression accurately. This step is an important part of the problem-solving process, as it helps to catch any potential errors and ensure the validity of the solution.

Conclusion

In conclusion, we have successfully simplified the expression ${\frac{\sqrt{12 x^3}}{\sqrt{3 x^2}}}$ to its simplest form, which is ${2 \sqrt{x}}$. This process involved breaking down the expression, simplifying the numerator and denominator separately, canceling common factors, and performing a final check to ensure accuracy. The key techniques used were factoring, applying the properties of square roots, and simplifying fractions. By systematically applying these methods, we were able to transform a complex-looking expression into a much more manageable one. This exercise underscores the importance of mastering algebraic simplification techniques, as they are fundamental to solving a wide range of mathematical problems. The ability to simplify expressions not only makes calculations easier but also enhances understanding and provides a clearer insight into the underlying relationships between variables and constants. Remember to always look for opportunities to simplify expressions whenever you encounter them, as it can often lead to a more elegant and efficient solution.

Final Answer: The final answer is ${\boxed{2 \sqrt{x}}}$ .