Which Expression Is Equivalent To 343 X 9 Y 12 Z 6 3 \sqrt[3]{343 X^9 Y^{12} Z^6} 3 343 X 9 Y 12 Z 6 ​ ?A. 7 X 3 Y 4 Z 2 7 X^3 Y^4 Z^2 7 X 3 Y 4 Z 2 B. 7 X 3 Y 6 Z 2 7 X^3 Y^6 Z^2 7 X 3 Y 6 Z 2 C. 49 X 3 Y 6 Z 2 49 X^3 Y^6 Z^2 49 X 3 Y 6 Z 2 D. 49 X 3 Y 4 Z 2 49 X^3 Y^4 Z^2 49 X 3 Y 4 Z 2

Introduction

Radical expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be broken down into more manageable parts. In this article, we will explore how to simplify radical expressions, focusing on the specific problem of finding an equivalent expression to $\sqrt[3]{343 x^9 y^{12} z^6}$.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol $\sqrt[3]{}$. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Breaking Down the Expression

To simplify the expression $\sqrt[3]{343 x^9 y^{12} z^6}$, we need to break it down into its prime factors. The number 343 can be factored as $7^3$, and the variables $x^9$, $y^{12}$, and $z^6$ can be broken down into their prime factors as well.

Prime Factorization

The prime factorization of 343 is $7^3$, which means that 343 can be expressed as the product of three 7s.

343 = 7^3

The prime factorization of $x^9$ is $x^3 \cdot x^3 \cdot x^3$, which means that $x^9$ can be expressed as the product of three $x^3$ terms.

x^9 = x^3 \cdot x^3 \cdot x^3

The prime factorization of $y^{12}$ is $y^4 \cdot y^4 \cdot y^4$, which means that $y^{12}$ can be expressed as the product of three $y^4$ terms.

y^{12} = y^4 \cdot y^4 \cdot y^4

The prime factorization of $z^6$ is $z^2 \cdot z^2 \cdot z^2$, which means that $z^6$ can be expressed as the product of three $z^2$ terms.

z^6 = z^2 \cdot z^2 \cdot z^2

Simplifying the Expression

Now that we have broken down the expression into its prime factors, we can simplify it by combining like terms.

\sqrt[3]{343 x^9 y^{12} z^6} = \sqrt[3]{7^3 \cdot x^3 \cdot x^3 \cdot x^3 \cdot y^4 \cdot y^4 \cdot y^4 \cdot z^2 \cdot z^2 \cdot z^2}

Using the properties of exponents, we can combine like terms and simplify the expression further.

\sqrt[3]{7^3 \cdot x^3 \cdot x^3 \cdot x^3 \cdot y^4 \cdot y^4 \cdot y^4 \cdot z^2 \cdot z^2 \cdot z^2} = 7 \cdot x^3 \cdot y^4 \cdot z^2

Conclusion

In conclusion, the expression $\sqrt[3]{343 x^9 y^{12} z^6}$ is equivalent to $7 x^3 y^4 z^2$. This can be verified by multiplying the expression by itself three times, which gives the original expression.

Answer

The correct answer is A. $7 x^3 y^4 z^2$.

Final Thoughts

Introduction

Radical expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be broken down into more manageable parts. In this article, we will explore some common questions and answers related to simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol $\sqrt[3]{}$. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break it down into its prime factors. The prime factorization of a number is the product of its prime factors. For example, the prime factorization of 343 is $7^3$, which means that 343 can be expressed as the product of three 7s.

Q: What is the difference between a cube root and a square root?

A: A cube root is a root that is raised to the power of 3, while a square root is a root that is raised to the power of 2. The cube root of a number is a value that, when multiplied by itself three times, gives the original number, while the square root of a number is a value that, when multiplied by itself two times, gives the original number.

Q: How do I simplify a radical expression with multiple variables?

A: To simplify a radical expression with multiple variables, you need to break it down into its prime factors and then combine like terms. For example, the expression $\sqrt[3]{343 x^9 y^{12} z^6}$ can be simplified by breaking it down into its prime factors and then combining like terms.

Q: What is the property of exponents that allows me to simplify radical expressions?

A: The property of exponents that allows you to simplify radical expressions is the product rule, which states that when you multiply two numbers with the same base, you can add their exponents. For example, $x^3 \cdot x^3 = x^{3+3} = x^6$.

Q: How do I know which radical expression is equivalent to the given expression?

A: To determine which radical expression is equivalent to the given expression, you need to simplify the expression by breaking it down into its prime factors and then combining like terms. For example, the expression $\sqrt[3]{343 x^9 y^{12} z^6}$ can be simplified to $7 x^3 y^4 z^2$.

Q: What is the final answer to the problem $\sqrt[3]{343 x^9 y^{12} z^6}$?

A: The final answer to the problem $\sqrt[3]{343 x^9 y^{12} z^6}$ is $7 x^3 y^4 z^2$.

Conclusion

In conclusion, simplifying radical expressions can be a challenging task, but with the right techniques and strategies, it can be broken down into more manageable parts. By understanding the properties of exponents and prime factorization, we can simplify complex expressions and arrive at the correct solution.

Common Mistakes to Avoid

• Not breaking down the expression into its prime factors
• Not combining like terms
• Not using the properties of exponents correctly
• Not checking the final answer

Tips and Tricks

• Use the properties of exponents to simplify radical expressions
• Break down the expression into its prime factors
• Combine like terms
• Check the final answer

Final Thoughts

Simplifying radical expressions can be a challenging task, but with the right techniques and strategies, it can be broken down into more manageable parts. By understanding the properties of exponents and prime factorization, we can simplify complex expressions and arrive at the correct solution.