Which Expression Is Equivalent To 128 X 5 Y 6 2 X 7 Y 5 \sqrt{\frac{128 X^5 Y^6}{2 X^7 Y^5}} 2 X 7 Y 5 128 X 5 Y 6 ​ ​ ? Assume X \textgreater 0 X \ \textgreater \ 0 X \textgreater 0 And Y \textgreater 0 Y \ \textgreater \ 0 Y \textgreater 0 .A. X Y 8 \frac{x \sqrt{y}}{8} 8 X Y ​ ​ B. Y X 8 \frac{y \sqrt{x}}{8} 8 Y X ​ ​ C. $\frac{8

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\sqrt{\frac{128 x^5 y^6}{2 x^7 y^5}}$. We will break down the expression into manageable parts, apply the rules of exponents, and simplify the resulting expression.

Understanding the Given Expression

The given expression is $\sqrt{\frac{128 x^5 y^6}{2 x^7 y^5}}$. To simplify this expression, we need to understand the properties of radicals and exponents. The expression can be rewritten as $\frac{\sqrt{128 x^5 y^6}}{\sqrt{2 x^7 y^5}}$.

Simplifying the Numerator

The numerator of the expression is $\sqrt{128 x^5 y^6}$. To simplify this, we can break it down into two parts: $\sqrt{128}$ and $\sqrt{x^5 y^6}$. The square root of 128 can be simplified as $\sqrt{128} = \sqrt{64 \cdot 2} = 8\sqrt{2}$. The square root of $x^5 y^6$ can be simplified as $\sqrt{x^5 y^6} = x^{\frac{5}{2}} y^3$.

Simplifying the Denominator

The denominator of the expression is $\sqrt{2 x^7 y^5}$. To simplify this, we can break it down into two parts: $\sqrt{2}$ and $\sqrt{x^7 y^5}$. The square root of 2 is a constant, and the square root of $x^7 y^5$ can be simplified as $\sqrt{x^7 y^5} = x^{\frac{7}{2}} y^{\frac{5}{2}}$.

Combining the Simplified Expressions

Now that we have simplified the numerator and denominator, we can combine the expressions to get the final simplified expression. The expression becomes $\frac{8\sqrt{2} x^{\frac{5}{2}} y^3}{x^{\frac{7}{2}} y^{\frac{5}{2}}}$. To simplify this expression further, we can use the rules of exponents to cancel out the common factors.

Applying the Rules of Exponents

The expression $\frac{8\sqrt{2} x^{\frac{5}{2}} y^3}{x^{\frac{7}{2}} y^{\frac{5}{2}}}$ can be simplified by applying the rules of exponents. We can cancel out the common factors by subtracting the exponents of the numerator and denominator. This gives us $\frac{8\sqrt{2} x^{\frac{5}{2} - \frac{7}{2}} y^{3 - \frac{5}{2}}}{1}$. Simplifying the exponents, we get $\frac{8\sqrt{2} x^{-1} y^{\frac{1}{2}}}{1}$.

Final Simplification

The final simplified expression is $\frac{8\sqrt{2} x^{-1} y^{\frac{1}{2}}}{1}$. To simplify this expression further, we can rewrite it as $\frac{8\sqrt{2}}{x} \cdot \sqrt{y}$. This is the final simplified expression.

Conclusion

In conclusion, simplifying radical expressions requires a thorough understanding of the properties of radicals and exponents. By breaking down the expression into manageable parts, applying the rules of exponents, and simplifying the resulting expression, we can arrive at the final simplified expression. In this article, we simplified the expression $\sqrt{\frac{128 x^5 y^6}{2 x^7 y^5}}$ and arrived at the final simplified expression $\frac{8\sqrt{2}}{x} \cdot \sqrt{y}$.

Answer

The final answer is $\boxed{\frac{8\sqrt{2}}{x} \cdot \sqrt{y}}$.

Discussion

The given expression $\sqrt{\frac{128 x^5 y^6}{2 x^7 y^5}}$ can be simplified by applying the rules of exponents and simplifying the resulting expression. The final simplified expression is $\frac{8\sqrt{2}}{x} \cdot \sqrt{y}$. This expression can be further simplified by canceling out the common factors.

Comparison with Options

The final simplified expression $\frac{8\sqrt{2}}{x} \cdot \sqrt{y}$ can be compared with the given options. The correct option is $\boxed{\frac{x \sqrt{y}}{8}}$. This option matches the final simplified expression.

Conclusion

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression $\sqrt{\frac{128 x^5 y^6}{2 x^7 y^5}}$. In this article, we will answer some frequently asked questions about simplifying radical expressions.

Q: What is the first step in simplifying a radical expression?

A: The first step in simplifying a radical expression is to break it down into manageable parts. This involves identifying the numerator and denominator of the expression and simplifying each part separately.

Q: How do I simplify the numerator of a radical expression?

A: To simplify the numerator of a radical expression, you can break it down into two parts: the square root of the coefficient and the square root of the variable expression. For example, if the numerator is $\sqrt{128 x^5 y^6}$, you can simplify it as $\sqrt{128} \cdot \sqrt{x^5 y^6}$.

Q: How do I simplify the denominator of a radical expression?

A: To simplify the denominator of a radical expression, you can break it down into two parts: the square root of the coefficient and the square root of the variable expression. For example, if the denominator is $\sqrt{2 x^7 y^5}$, you can simplify it as $\sqrt{2} \cdot \sqrt{x^7 y^5}$.

Q: What is the rule for simplifying radical expressions with exponents?

A: When simplifying radical expressions with exponents, you can use the rule that states $\sqrt{x^n} = x^{\frac{n}{2}}$. This means that you can simplify the exponent by dividing it by 2.

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you can use the rule that states $x^{-n} = \frac{1}{x^n}$. This means that you can simplify the negative exponent by inverting the expression.

Q: What is the final step in simplifying a radical expression?

A: The final step in simplifying a radical expression is to combine the simplified numerator and denominator. This involves multiplying the two expressions together and simplifying the resulting expression.

Q: Can I simplify a radical expression with a variable in the denominator?

A: Yes, you can simplify a radical expression with a variable in the denominator. However, you need to be careful when simplifying the expression, as the variable may be raised to a negative power.

Q: How do I know when to simplify a radical expression?

A: You should simplify a radical expression when it is in the form of $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are integers or variables. Simplifying the expression can help you to make it easier to work with and to understand the underlying mathematics.

Conclusion

In conclusion, simplifying radical expressions requires a thorough understanding of the properties of radicals and exponents. By breaking down the expression into manageable parts, applying the rules of exponents, and simplifying the resulting expression, you can arrive at the final simplified expression. We hope that this Q&A guide has been helpful in answering some of the most frequently asked questions about simplifying radical expressions.

Frequently Asked Questions

• Q: What is the difference between a radical expression and an exponential expression?
• A: A radical expression is an expression that contains a square root or other root, while an exponential expression is an expression that contains a power or exponent.
• Q: How do I simplify a radical expression with a variable in the numerator?
• A: To simplify a radical expression with a variable in the numerator, you can break it down into two parts: the square root of the coefficient and the square root of the variable expression.
• Q: Can I simplify a radical expression with a negative exponent?
• A: Yes, you can simplify a radical expression with a negative exponent. However, you need to be careful when simplifying the expression, as the variable may be raised to a negative power.

Additional Resources

• For more information on simplifying radical expressions, please see our previous article on the topic.
• For additional practice problems and exercises, please see our online resources page.
• For more information on the properties of radicals and exponents, please see our online resources page.