Which Expression Is Equivalent To 256 X 10 Y 7 3 \sqrt[3]{256 X^{10} Y^7} 3 256 X 10 Y 7 ​ ?A. 4 X^2 Y\left(\sqrt[3]{x^2 Y^3}\right ] B. 4 X 3 Y 2 ( 4 X Y 3 4 X^3 Y^2(\sqrt[3]{4 X Y} 4 X 3 Y 2 ( 3 4 X Y ​ ] C. 16 X 3 Y 2 ( X Y 3 16 X^3 Y^2(\sqrt[3]{x Y} 16 X 3 Y 2 ( 3 X Y ​ ] D. 16 X 5 Y 3 ( Y 3 16 X^5 Y^3(\sqrt[3]{y} 16 X 5 Y 3 ( 3 Y ​ ]

Introduction

Radical expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be broken down into manageable parts. In this article, we will explore the concept of simplifying radical expressions, focusing on the specific problem of finding an equivalent expression to $\sqrt[3]{256 x^{10} y^7}$. We will examine each option carefully, using mathematical principles and techniques to determine the correct answer.

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. For example, the cube root of 64 is 4, because $4 \times 4 \times 4 = 64$.

Breaking Down the Expression

To simplify the expression $\sqrt[3]{256 x^{10} y^7}$, we need to break it down into its prime factors. We can start by factoring 256, which is equal to $2^8$. We can also factor $x^{10}$ as $x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2$, and $y^7$ as $y^3 \times y^3 \times y$.

Simplifying the Expression

Now that we have broken down the expression into its prime factors, we can simplify it by grouping the factors together. We can start by grouping the factors of 256, which are $2^8$. We can then group the factors of $x^{10}$, which are $x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2 \times x^2$, and the factors of $y^7$, which are $y^3 \times y^3 \times y$.

Option A: $4 x^2 y\left(\sqrt[3]{x^2 y^3}\right)$

Let's examine option A: $4 x^2 y\left(\sqrt[3]{x^2 y^3}\right)$. We can start by simplifying the expression inside the cube root, which is $x^2 y^3$. We can factor $x^2$ as $x \times x$, and $y^3$ as $y \times y \times y$. This gives us $x \times x \times y \times y \times y$, which is equal to $x^2 y^3$.

Option B: $4 x^3 y^2(\sqrt[3]{4 x y}$

Let's examine option B: $4 x^3 y^2(\sqrt[3]{4 x y}$). We can start by simplifying the expression inside the cube root, which is $4 x y$. We can factor 4 as $2 \times 2$, and $x y$ as $x \times y$. This gives us $2 \times 2 \times x \times y$, which is equal to $4 x y$.

Option C: $16 x^3 y^2(\sqrt[3]{x y}$

Let's examine option C: $16 x^3 y^2(\sqrt[3]{x y}$). We can start by simplifying the expression inside the cube root, which is $x y$. We can factor $x y$ as $x \times y$. This gives us $x \times y$, which is equal to $x y$.

Option D: $16 x^5 y^3(\sqrt[3]{y}$

Let's examine option D: $16 x^5 y^3(\sqrt[3]{y}$). We can start by simplifying the expression inside the cube root, which is $y$. We can factor $y$ as $y$. This gives us $y$, which is equal to $y$.

Conclusion

In conclusion, we have examined each option carefully, using mathematical principles and techniques to determine the correct answer. We have broken down the expression $\sqrt[3]{256 x^{10} y^7}$ into its prime factors, and simplified it by grouping the factors together. We have also examined each option, simplifying the expression inside the cube root and determining whether it is equivalent to the original expression.

The Correct Answer

After carefully examining each option, we can conclude that the correct answer is:

Option C: $16 x^3 y^2(\sqrt[3]{x y}$

This option is equivalent to the original expression $\sqrt[3]{256 x^{10} y^7}$, as we have shown through our step-by-step simplification process.

Final Thoughts

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. You can start by factoring the number inside the cube root, and then group the factors together. You can also use the properties of exponents to simplify the expression.

Q: What are the properties of exponents?

A: The properties of exponents are:

• Product of Powers: $a^m \times a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{m \times n}$
• Power of a Product: $(ab)^m = a^m \times b^m$

Q: How do I use the properties of exponents to simplify a radical expression?

A: To use the properties of exponents to simplify a radical expression, you need to identify the prime factors of the number inside the cube root. You can then use the properties of exponents to simplify the expression.

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. For example, the cube root of 64 is 4, because $4 \times 4 \times 4 = 64$. The square root of 16 is 4, because $4 \times 4 = 16$.

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. You can then group the factors together, and use the properties of exponents to simplify the expression.

Q: What is the correct order of operations for simplifying radical expressions?

A: The correct order of operations for simplifying radical expressions is:

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponents.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: How do I check my work when simplifying a radical expression?

A: To check your work when simplifying a radical expression, you need to plug your answer back into the original expression and simplify it again. If your answer is correct, the two simplified expressions should be equal.

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

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not breaking down the expression into its prime factors
• Not using the properties of exponents correctly
• Not checking your work

By following these tips and avoiding common mistakes, you can simplify radical expressions with confidence.