Which Expression Is Equivalent To $\sqrt[3]{x^5 Y}$?A. $x^{\frac{5}{3}} Y$B. $x^{\frac{5}{3}} Y^{\frac{1}{3}}$C. $x^{\frac{3}{5}} Y$D. $x^{\frac{3}{5}} Y^3$

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Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression x5y3\sqrt[3]{x^5 y} and explore the different equivalent expressions that can be obtained.

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, which is denoted by the symbol 3\sqrt[3]{}. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Simplifying the Expression

To simplify the expression x5y3\sqrt[3]{x^5 y}, we need to apply the properties of exponents and radicals. We can start by breaking down the expression into its individual components:

x5y3=x53â‹…y3\sqrt[3]{x^5 y} = \sqrt[3]{x^5} \cdot \sqrt[3]{y}

Now, we can apply the property of exponents that states ann=a\sqrt[n]{a^n} = a. In this case, we have:

x53=x53\sqrt[3]{x^5} = x^{\frac{5}{3}}

So, the expression becomes:

x5y3=x53â‹…y3\sqrt[3]{x^5 y} = x^{\frac{5}{3}} \cdot \sqrt[3]{y}

Finding the Equivalent Expression

Now, we need to find the equivalent expression for y3\sqrt[3]{y}. We can do this by applying the property of radicals that states an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}. In this case, we have:

y3=y13\sqrt[3]{y} = y^{\frac{1}{3}}

So, the expression becomes:

x5y3=x53y13\sqrt[3]{x^5 y} = x^{\frac{5}{3}} y^{\frac{1}{3}}

Comparing the Options

Now, let's compare the simplified expression with the options provided:

A. x53yx^{\frac{5}{3}} y B. x53y13x^{\frac{5}{3}} y^{\frac{1}{3}} C. x35yx^{\frac{3}{5}} y D. x35y3x^{\frac{3}{5}} y^3

We can see that option B is the only one that matches the simplified expression.

Conclusion

In conclusion, the expression x5y3\sqrt[3]{x^5 y} is equivalent to x53y13x^{\frac{5}{3}} y^{\frac{1}{3}}. This can be obtained by applying the properties of exponents and radicals, and by simplifying the expression using the rules of algebra.

Key Takeaways

  • Radical expressions can be simplified using the properties of exponents and radicals.
  • The cube root of a number can be expressed as a power of the number.
  • The expression x5y3\sqrt[3]{x^5 y} is equivalent to x53y13x^{\frac{5}{3}} y^{\frac{1}{3}}.

Frequently Asked Questions

Q: What is the cube root of a number?

A: 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 apply the properties of exponents and radicals, and use the rules of algebra to simplify the expression.

Q: What is the equivalent expression for x5y3\sqrt[3]{x^5 y}?

A: The equivalent expression for x5y3\sqrt[3]{x^5 y} is x53y13x^{\frac{5}{3}} y^{\frac{1}{3}}.

References