Simplify Radicals: Find A, B, And C

Hey math enthusiasts! Let's dive into a fun problem involving radicals. We're going to simplify a radical expression and figure out the values of a, b, and c that make it all work. This is a classic example of how to break down complex expressions into simpler forms. Ready to get started? Let's go!

Understanding the Problem: Simplifying Radicals

So, the question is: find the values for a, b, and c that complete the simplification of the radical expression. We're given an expression involving variables x, y, and z under a square root, and we need to simplify it. The goal is to rewrite the expression in a way that pulls out as many terms as possible from under the radical sign. This process involves using the properties of radicals and exponents to rewrite the expression in a more manageable form. Specifically, we'll use the rule that the square root of a product is the product of the square roots. We will break down the original radical expression step by step, utilizing the properties of exponents and radicals to arrive at the simplified form. This will make it easier to identify the values of a, b, and c. The core idea is to separate out perfect squares from the terms under the radical, since the square root of a perfect square is a whole number or a simpler expression. This process is fundamental in algebra and is used extensively in solving various types of equations and simplifying complex mathematical expressions. It's all about making the expression as clear and concise as possible.

The original expression is $\sqrt{x^{12} y^9 z^5}$. We're also given a hint: $\sqrt{x^{12} y^9 z^5}=\sqrt{x^{12} \cdot y^8 \cdot y \cdot z^4 \cdot z}=x^a y^b z^c \sqrt{y z}$. Our job is to figure out what a, b, and c are. The given hint provides a roadmap for simplifying the expression by breaking down the terms under the square root into their component parts. The breakdown involves rewriting the expression so that perfect squares can be easily identified and extracted from the radical. This is a common strategy when working with radicals, and it simplifies the process by enabling us to isolate whole-number or simpler expressions outside the radical. To do this, we rewrite the powers of x, y, and z so that the exponents can be divided by 2 (since we are taking a square root). The terms with exponents that are divisible by 2 will come out of the square root, and the remaining terms will stay inside. This approach ensures that we systematically reduce the complexity of the radical expression.

Now, let's break down the given equation and figure out those values for a, b, and c. It's like a puzzle, and we're going to solve it together, step by step. This process not only allows us to find the specific values of a, b, and c, but it also reinforces our understanding of the properties of radicals and exponents. As we work through the simplification, we'll see how each step contributes to the overall solution. The skill of simplifying radicals is crucial in higher-level math courses and is used extensively in fields like physics and engineering. So, let's get into it, and you'll find that it's actually quite fun!

Step-by-Step Solution: Unveiling a, b, and c

Let's get down to business! Here's how we find the values of a, b, and c. We start with the expression: $\sqrt{x^{12} y^9 z^5}$.

1. Breaking it down: The first step involves breaking down the expression under the square root into terms that we can simplify. Remember, the goal is to rewrite the terms so that we can easily identify and extract perfect squares. We'll leverage the property that the square root of a product is the product of the square roots to help us. This property is key because it allows us to break down a complex radical into simpler terms.

   We can rewrite $y^9$ as $y^8 \cdot y$ because $y^8$ is a perfect square ($y^8 = (y^4)^2$) and $y$ remains under the radical. Similarly, we rewrite $z^5$ as $z^4 \cdot z$, where $z^4$ is a perfect square ($z^4 = (z^2)^2$) and $z$ remains under the radical. The term $x^{12}$ is already a perfect square, as $x^{12} = (x^6)^2$. The idea is to isolate terms that can be expressed as a perfect square, as this will help us extract them from the square root. By using this method, we can manipulate the original expression into a form that's easier to solve.

   So, we get: $\sqrt{x^{12} \cdot y^8 \cdot y \cdot z^4 \cdot z}$.

2. Taking the square root: Now we take the square root of each of the perfect squares. The square root of $x^{12}$ is $x^6$ (since $12 / 2 = 6$). The square root of $y^8$ is $y^4$ (since $8 / 2 = 4$). The square root of $z^4$ is $z^2$ (since $4 / 2 = 2$).

   This means we can rewrite the expression as: $x^6 \cdot y^4 \cdot z^2 \cdot \sqrt{y z}$.

3. Comparing and solving for a, b, and c: Now we compare this result to $x^a y^b z^c \sqrt{y z}$.

   • For the x term, we have $x^6$, so a = 6.
   • For the y term, we have $y^4$, so b = 4.
   • For the z term, we have $z^2$, so c = 2.

   Therefore, the values are a = 6, b = 4, and c = 2.

Final Answer and Conclusion: The Simplification Complete

And there you have it, guys! We've successfully simplified the radical expression and found the values for a, b, and c. It's always a good feeling when we solve these math puzzles, right? This process reinforces the important principles of exponents and radicals. By understanding how to break down complex expressions into simpler forms, we're better equipped to handle a wide range of mathematical problems. Remember, the key is to look for perfect squares within the radicals and apply the properties of exponents.

So, to recap:

• $a = 6$
• $b = 4$
• $c = 2$

This example is a great demonstration of how important it is to master basic mathematical concepts. These concepts form the building blocks for more advanced topics in algebra and beyond. Keep practicing, and you'll find that these problems become easier over time. Understanding these concepts is essential not just for academic success but also for everyday problem-solving. It's a testament to how math can be both interesting and useful.

Keep up the great work, and happy simplifying!