Solving The Algebraic Expression: A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of algebra to tackle a fascinating problem. We're going to break down the expression $\sqrt{ \frac{1}{16x^2a^4} } \times 3 \times \sqrt{ \frac{9x^4}{18} }$ step by step, making sure everyone understands the process. Whether you're an algebra whiz or just starting out, this guide is for you. Let's get started and unlock the secrets of this algebraic puzzle!

Understanding the Expression

First, let's take a good look at the expression: $\sqrt{ \frac{1}{16x^2a^4} } \times 3 \times \sqrt{ \frac{9x^4}{18} }$. It might seem a bit intimidating at first, but don't worry, we'll break it down into manageable parts. The expression involves square roots, fractions, and variables, all common elements in algebra. Our goal is to simplify this expression, which means we want to make it as neat and concise as possible. We'll do this by applying the rules of algebra and simplifying each part of the expression systematically.

When we see square roots, we should think about how we can simplify them. 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 times 3 is 9. In our expression, we have square roots of fractions, which might seem tricky, but we can handle them by taking the square root of the numerator (the top part of the fraction) and the square root of the denominator (the bottom part of the fraction) separately. This is a key strategy we'll use to simplify the expression. We'll also need to keep in mind the properties of exponents, as we have variables raised to powers. For instance, $x^2$ means x multiplied by itself, and we'll need to deal with these exponents carefully when we simplify the square roots.

Algebraic expressions can often look complex, but they are built on fundamental rules and principles. By understanding these principles, we can tackle any expression with confidence. In this case, the key is to simplify the square roots and fractions one step at a time. We'll use the properties of square roots to separate the numerators and denominators, and we'll simplify the fractions by canceling out common factors. This step-by-step approach will make the whole process much easier and less overwhelming. So, let's roll up our sleeves and start simplifying!

Step-by-Step Solution

Let's break this down step by step. The expression we're working with is: $\sqrt{ \frac{1}{16x^2a^4} } \times 3 \times \sqrt{ \frac{9x^4}{18} }$.

Step 1: Simplify the First Square Root

The first part we'll tackle is $\sqrt{ \frac{1}{16x^2a^4} }$. To simplify this, we can take the square root of the numerator and the denominator separately. The square root of 1 is simply 1. For the denominator, we have $16x^2a^4$. The square root of 16 is 4, the square root of $x^2$ is x (assuming x is non-negative), and the square root of $a^4$ is $a^2$. So, $\sqrt{16x^2a^4}$ simplifies to $4xa^2$. Therefore, $\sqrt{ \frac{1}{16x^2a^4} }$ becomes $\frac{1}{4xa^2}$.

Step 2: Simplify the Second Square Root

Now, let's move on to the second square root: $\sqrt{ \frac{9x^4}{18} }$. Before we take the square root, we can simplify the fraction inside. We have $\frac{9x^4}{18}$. Both 9 and 18 are divisible by 9, so we can simplify the fraction to $\frac{x^4}{2}$. Now we have $\sqrt{ \frac{x^4}{2} }$. The square root of $x^4$ is $x^2$. So, we have $\frac{x^2}{\sqrt{2}}$. To rationalize the denominator (get rid of the square root in the denominator), we multiply both the numerator and the denominator by $\sqrt{2}$. This gives us $\frac{x^2\sqrt{2}}{2}$.

Step 3: Combine and Simplify

Now we have: $\frac{1}{4xa^2} \times 3 \times \frac{x^2\sqrt{2}}{2}$. Let's multiply these together. We get $\frac{3x^2\sqrt{2}}{8xa^2}$. We can simplify this further by canceling out an x from the numerator and the denominator. This leaves us with $\frac{3x\sqrt{2}}{8a^2}$.

Simplifying algebraic expressions like this involves a combination of steps, each building on the previous one. The key is to break down the expression into smaller, more manageable parts. By simplifying each square root and fraction separately, we can avoid getting overwhelmed by the complexity of the entire expression. Rationalizing the denominator, as we did in step 2, is a common technique in algebra to make expressions neater and easier to work with. Finally, combining the simplified parts and canceling out common factors leads us to the final, simplified form of the expression. This step-by-step process not only helps us solve the problem but also enhances our understanding of algebraic principles.

Final Answer

So, after all the simplification, our final answer is: $\frac{3x\sqrt{2}}{8a^2}$.

Let's Recap

To recap, we started with a complex expression involving square roots, fractions, and variables. We broke it down into smaller, more manageable parts. We simplified each square root separately, rationalized the denominator in one part, and then combined everything. By following these steps, we were able to simplify the expression and arrive at our final answer. Remember, algebra is all about breaking down problems and solving them step by step. Don't be intimidated by complex expressions – just take it one step at a time, and you'll get there!

Key Takeaways

Here are some key takeaways from this problem:

1. Simplify Square Roots: When you see a square root, try to simplify it by finding the square roots of the numerator and denominator separately.
2. Simplify Fractions: Before doing anything else, simplify the fractions inside the square roots or elsewhere in the expression.
3. Rationalize Denominators: If you have a square root in the denominator, rationalize it by multiplying both the numerator and denominator by the square root.
4. Combine and Simplify: Once you've simplified the individual parts, combine them and look for common factors that can be canceled out.

Algebra is a powerful tool, and mastering these techniques will help you solve a wide range of problems. Practice is key, so keep working on different expressions and problems. The more you practice, the more confident you'll become in your algebraic skills. And remember, every complex problem can be broken down into simpler steps. So, keep learning, keep practicing, and keep exploring the fascinating world of algebra!

Tips for Solving Similar Problems

When you encounter similar problems, here are some tips and tricks to keep in mind. First off, always look for opportunities to simplify. This might mean simplifying fractions, taking square roots, or combining like terms. The more you simplify early on, the easier the rest of the problem will be. Secondly, pay close attention to the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following the correct order is crucial for getting the right answer. Thirdly, don't be afraid to break the problem down into smaller steps. This is especially helpful when dealing with complex expressions. By breaking the problem down, you can focus on each part individually and avoid making mistakes.

Algebraic manipulation is a skill that improves with practice. The more problems you solve, the better you'll become at recognizing patterns and applying the correct techniques. One of the best ways to practice is to work through a variety of problems, from simple to complex. Look for problems in textbooks, online resources, or worksheets. When you solve a problem, take the time to understand each step and why it's necessary. This will help you build a deeper understanding of the concepts and make it easier to solve similar problems in the future. And if you get stuck, don't be afraid to ask for help. There are many resources available, including teachers, tutors, and online forums.

Finally, remember that algebra is not just about memorizing formulas and procedures. It's about developing problem-solving skills and logical thinking. When you approach a problem, think about the underlying concepts and how they apply to the situation. Ask yourself questions like,