Find The Values For { A, B,$}$ And { C$}$ That Complete The Simplification.$[ \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

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Introduction

Radicals are an essential part of mathematics, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of radicals and explore how to simplify them using the properties of exponents. We will focus on finding the values for $a$, $b$, and $c$ that complete the simplification of the given radical expression.

Understanding Radicals

A radical is a mathematical expression that involves a root or a power of a number. It is denoted by the symbol $\sqrt{}$ and is used to represent the square root of a number. For example, $\sqrt{16}$ represents the square root of 16, which is equal to 4.

Simplifying Radicals using Exponents

When simplifying radicals, we can use the properties of exponents to rewrite the expression in a simpler form. One of the key properties of exponents is that when we multiply two numbers with the same base, we can add their exponents. For example, $x^a \cdot x^b = x^{a+b}$.

The Given Radical Expression

The given radical expression is $\sqrt{x^{12} y^9 z^5}$. Our goal is to simplify this expression using the properties of exponents and find the values for $a$, $b$, and $c$ that complete the simplification.

Breaking Down the Radical Expression

To simplify the radical expression, we can break it down into smaller parts. We can rewrite the expression as $\sqrt{x^{12} \cdot y^8 \cdot y \cdot z^4 \cdot z}$.

Using the Properties of Exponents

Now that we have broken down the radical expression, we can use the properties of exponents to simplify it further. We can rewrite the expression as $x^a y^b z^c \sqrt{y}$.

Finding the Values for $a$, $b$, and $c$

To find the values for $a$, $b$, and $c$, we need to analyze the exponents of $x$, $y$, and $z$ in the simplified expression. We can see that the exponent of $x$ is $a$, the exponent of $y$ is $b$, and the exponent of $z$ is $c$.

Simplifying the Exponents

We can simplify the exponents by using the properties of exponents. We can rewrite the expression as $x^{12} y^8 z^4 \sqrt{y}$.

Finding the Values for $a$, $b$, and $c$

Now that we have simplified the exponents, we can find the values for $a$, $b$, and $c$. We can see that $a = 12$, $b = 8 + \frac{1}{2}$, and $c = 4 + \frac{1}{2}$.

Conclusion

In this article, we have simplified the radical expression $\sqrt{x^{12} y^9 z^5}$ using the properties of exponents. We have found the values for $a$, $b$, and $c$ that complete the simplification. We have also analyzed the exponents of $x$, $y$, and $z$ in the simplified expression and simplified them using the properties of exponents.

Final Answer

The final answer is $a = 12$, $b = 8 + \frac{1}{2}$, and $c = 4 + \frac{1}{2}$.

Frequently Asked Questions

Q: What is the given radical expression?

A: The given radical expression is $\sqrt{x^{12} y^9 z^5}$.

Q: How do we simplify the radical expression?

A: We can simplify the radical expression by breaking it down into smaller parts and using the properties of exponents.

Q: What are the values for $a$, $b$, and $c$?

A: The values for $a$, $b$, and $c$ are $a = 12$, $b = 8 + \frac{1}{2}$, and $c = 4 + \frac{1}{2}$.

Q: How do we find the values for $a$, $b$, and $c$?

A: We can find the values for $a$, $b$, and $c$ by analyzing the exponents of $x$, $y$, and $z$ in the simplified expression and simplifying them using the properties of exponents.

Step-by-Step Solution

Step 1: Break down the radical expression

We can rewrite the expression as $\sqrt{x^{12} \cdot y^8 \cdot y \cdot z^4 \cdot z}$.

Step 2: Use the properties of exponents

We can rewrite the expression as $x^a y^b z^c \sqrt{y}$.

Step 3: Simplify the exponents

We can rewrite the expression as $x^{12} y^8 z^4 \sqrt{y}$.

Step 4: Find the values for $a$, $b$, and $c$

We can see that $a = 12$, $b = 8 + \frac{1}{2}$, and $c = 4 + \frac{1}{2}$.

Common Mistakes

Mistake 1: Not breaking down the radical expression

Not breaking down the radical expression can make it difficult to simplify and find the values for $a$, $b$, and $c$.

Mistake 2: Not using the properties of exponents

Not using the properties of exponents can make it difficult to simplify the exponents and find the values for $a$, $b$, and $c$.

Mistake 3: Not analyzing the exponents

Not analyzing the exponents of $x$, $y$, and $z$ in the simplified expression can make it difficult to find the values for $a$, $b$, and $c$.

Real-World Applications

Simplifying Radicals in Algebra

Simplifying radicals is an essential skill in algebra, and it is used to solve equations and inequalities.

Simplifying Radicals in Geometry

Simplifying radicals is also used in geometry to find the lengths of sides and the areas of shapes.

Simplifying Radicals in Calculus

Simplifying radicals is used in calculus to find the derivatives and integrals of functions.

Conclusion

In conclusion, simplifying radicals is an essential skill in mathematics, and it is used to solve equations and inequalities, find the lengths of sides and the areas of shapes, and find the derivatives and integrals of functions. We have simplified the radical expression $\sqrt{x^{12} y^9 z^5}$ using the properties of exponents and found the values for $a$, $b$, and $c$ that complete the simplification.

Introduction

In our previous article, we explored the concept of simplifying radicals using the properties of exponents. We broke down the radical expression $\sqrt{x^{12} y^9 z^5}$ and found the values for $a$, $b$, and $c$ that complete the simplification. In this article, we will answer some of the most frequently asked questions about simplifying radicals.

Q: What is the difference between a radical and an exponent?

A: A radical is a mathematical expression that involves a root or a power of a number, while an exponent is a number that represents the power to which a base number is raised.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can break it down into smaller parts and use the properties of exponents to rewrite the expression in a simpler form.

Q: What are the properties of exponents that I can use to simplify radicals?

A: The properties of exponents that you can use to simplify radicals include:

• When multiplying two numbers with the same base, you can add their exponents.
• When dividing two numbers with the same base, you can subtract their exponents.
• When raising a power to a power, you can multiply the exponents.

Q: How do I find the values for $a$, $b$, and $c$ in a simplified radical expression?

A: To find the values for $a$, $b$, and $c$ in a simplified radical expression, you can analyze the exponents of $x$, $y$, and $z$ in the simplified expression and simplify them using the properties of exponents.

Q: What are some common mistakes to avoid when simplifying radicals?

A: Some common mistakes to avoid when simplifying radicals include:

• Not breaking down the radical expression into smaller parts.
• Not using the properties of exponents to rewrite the expression in a simpler form.
• Not analyzing the exponents of $x$, $y$, and $z$ in the simplified expression.

Q: How do I apply simplifying radicals in real-world situations?

A: Simplifying radicals is used in a variety of real-world situations, including:

• Algebra: Simplifying radicals is used to solve equations and inequalities.
• Geometry: Simplifying radicals is used to find the lengths of sides and the areas of shapes.
• Calculus: Simplifying radicals is used to find the derivatives and integrals of functions.

Q: What are some tips for simplifying radicals?

A: Some tips for simplifying radicals include:

• Start by breaking down the radical expression into smaller parts.
• Use the properties of exponents to rewrite the expression in a simpler form.
• Analyze the exponents of $x$, $y$, and $z$ in the simplified expression.
• Check your work to make sure that the simplified expression is correct.

Q: How do I know if I have simplified a radical expression correctly?

A: To check if you have simplified a radical expression correctly, you can:

• Verify that the simplified expression is in the correct form.
• Check that the exponents of $x$, $y$, and $z$ are correct.
• Make sure that the simplified expression is consistent with the original expression.

Q: What are some common applications of simplifying radicals?

A: Some common applications of simplifying radicals include:

• Finding the lengths of sides and the areas of shapes in geometry.
• Solving equations and inequalities in algebra.
• Finding the derivatives and integrals of functions in calculus.

Q: How do I use simplifying radicals to solve equations and inequalities?

A: To use simplifying radicals to solve equations and inequalities, you can:

• Start by simplifying the radical expression on both sides of the equation or inequality.
• Use the properties of exponents to rewrite the expression in a simpler form.
• Analyze the exponents of $x$, $y$, and $z$ in the simplified expression.
• Check your work to make sure that the solution is correct.

Q: What are some common mistakes to avoid when using simplifying radicals to solve equations and inequalities?

A: Some common mistakes to avoid when using simplifying radicals to solve equations and inequalities include:

• Not simplifying the radical expression on both sides of the equation or inequality.
• Not using the properties of exponents to rewrite the expression in a simpler form.
• Not analyzing the exponents of $x$, $y$, and $z$ in the simplified expression.

Q: How do I use simplifying radicals to find the lengths of sides and the areas of shapes in geometry?

A: To use simplifying radicals to find the lengths of sides and the areas of shapes in geometry, you can:

• Start by simplifying the radical expression that represents the length of the side or the area of the shape.
• Use the properties of exponents to rewrite the expression in a simpler form.
• Analyze the exponents of $x$, $y$, and $z$ in the simplified expression.
• Check your work to make sure that the length of the side or the area of the shape is correct.

Q: What are some common mistakes to avoid when using simplifying radicals to find the lengths of sides and the areas of shapes in geometry?

A: Some common mistakes to avoid when using simplifying radicals to find the lengths of sides and the areas of shapes in geometry include:

• Not simplifying the radical expression that represents the length of the side or the area of the shape.
• Not using the properties of exponents to rewrite the expression in a simpler form.
• Not analyzing the exponents of $x$, $y$, and $z$ in the simplified expression.

Q: How do I use simplifying radicals to find the derivatives and integrals of functions in calculus?

A: To use simplifying radicals to find the derivatives and integrals of functions in calculus, you can:

• Start by simplifying the radical expression that represents the derivative or the integral of the function.
• Use the properties of exponents to rewrite the expression in a simpler form.
• Analyze the exponents of $x$, $y$, and $z$ in the simplified expression.
• Check your work to make sure that the derivative or the integral of the function is correct.

Q: What are some common mistakes to avoid when using simplifying radicals to find the derivatives and integrals of functions in calculus?

A: Some common mistakes to avoid when using simplifying radicals to find the derivatives and integrals of functions in calculus include:

• Not simplifying the radical expression that represents the derivative or the integral of the function.
• Not using the properties of exponents to rewrite the expression in a simpler form.
• Not analyzing the exponents of $x$, $y$, and $z$ in the simplified expression.

Conclusion

In conclusion, simplifying radicals is an essential skill in mathematics, and it is used to solve equations and inequalities, find the lengths of sides and the areas of shapes, and find the derivatives and integrals of functions. We have answered some of the most frequently asked questions about simplifying radicals and provided tips and examples to help you master this skill.