What Is The Simplified Form Of The Following Expression?A. 4 X 5 3 \sqrt[3]{\frac{4x}{5}} 3 5 4 X ​ ​ B. 4 X 3 5 \frac{\sqrt[3]{4x}}{5} 5 3 4 X ​ ​ C. 20 X 3 5 \frac{\sqrt[3]{20x}}{5} 5 3 20 X ​ ​ D. 100 X 3 5 \frac{\sqrt[3]{100x}}{5} 5 3 100 X ​ ​ E. 100 X 3 125 \frac{\sqrt[3]{100x}}{125} 125 3 100 X ​ ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the simplified form of the expression $\sqrt[3]{\frac{4x}{5}}$ and examine the various options provided.

Understanding the Expression

The given expression is $\sqrt[3]{\frac{4x}{5}}$. To simplify this expression, we need to understand the properties of radicals and exponents. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In this case, we have a fraction under the cube root, which can be simplified using the properties of radicals.

Simplifying the Expression

To simplify the expression, we can start by factoring the numerator and denominator. The numerator, $4x$, can be factored as $4x = 4 \cdot x$. The denominator, $5$, is already factored.

\sqrt[3]{\frac{4x}{5}} = \sqrt[3]{\frac{4 \cdot x}{5}}

Next, we can use the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Applying this property to the expression, we get:

\sqrt[3]{\frac{4 \cdot x}{5}} = \sqrt[3]{4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{5}}

Now, we can simplify each radical individually. The cube root of $4$ is $2$, the cube root of $x$ is still $x$, and the cube root of $\frac{1}{5}$ is $\frac{1}{\sqrt[3]{5}}$.

\sqrt[3]{4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{5}} = 2 \cdot x \cdot \frac{1}{\sqrt[3]{5}}

Finally, we can simplify the expression by multiplying the terms together.

2 \cdot x \cdot \frac{1}{\sqrt[3]{5}} = \frac{2x}{\sqrt[3]{5}}

Comparing with the Options

Now that we have simplified the expression, we can compare it with the options provided.

• Option A: $\sqrt[3]{\frac{4x}{5}}$
• Option B: $\frac{\sqrt[3]{4x}}{5}$
• Option C: $\frac{\sqrt[3]{20x}}{5}$
• Option D: $\frac{\sqrt[3]{100x}}{5}$
• Option E: $\frac{\sqrt[3]{100x}}{125}$

Our simplified expression, $\frac{2x}{\sqrt[3]{5}}$, is not among the options. However, we can rewrite it in a form that matches one of the options.

\frac{2x}{\sqrt[3]{5}} = \frac{2x}{\sqrt[3]{5}} \cdot \frac{\sqrt[3]{5^2}}{\sqrt[3]{5^2}} = \frac{2x\sqrt[3]{5^2}}{\sqrt[3]{5^3}} = \frac{2x\sqrt[3]{25}}{5}

This expression is similar to option C, but it is not exactly the same. However, we can simplify it further by recognizing that $\sqrt[3]{25} = \sqrt[3]{5^2} = 5^{\frac{2}{3}}$.

\frac{2x\sqrt[3]{25}}{5} = \frac{2x \cdot 5^{\frac{2}{3}}}{5} = \frac{2x \cdot 5^{\frac{2}{3}}}{5} \cdot \frac{5^{\frac{1}{3}}}{5^{\frac{1}{3}}} = \frac{2x \cdot 5^{\frac{3}{3}}}{5^{\frac{4}{3}}} = \frac{2x \cdot 5}{5^{\frac{4}{3}}} = \frac{10x}{5^{\frac{4}{3}}} = \frac{10x}{5^{\frac{4}{3}}} \cdot \frac{5^{\frac{4}{3}}}{5^{\frac{4}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{5^{\frac{8}{3}}} = \frac{10x \cdot 5^{\frac{4}{3}}}{<br/>
**Simplifying Radical Expressions: A Q&A Guide**
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Q: What is the simplified form of the expression $\sqrt[3]{\frac{4x}{5}}$?
A: The simplified form of the expression $\sqrt[3]{\frac{4x}{5}}$ is $\frac{2x}{\sqrt[3]{5}}$.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you can start by factoring the numerator and denominator. Then, use the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Finally, simplify each radical individually.
Q: What is the difference between a radical and an exponent?
A: A radical is a symbol that represents the nth root of a number, while an exponent is a small number that is raised to a power. For example, $\sqrt[3]{x}$ is a radical, while $x^3$ is an exponent.
Q: Can I simplify a radical expression by canceling out common factors?
A: Yes, you can simplify a radical expression by canceling out common factors. For example, $\sqrt[3]{\frac{4x}{5}}$ can be simplified by canceling out the common factor of $x$.
Q: How do I simplify a radical expression with a fraction under the radical?
A: To simplify a radical expression with a fraction under the radical, you can start by factoring the numerator and denominator. Then, use the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Finally, simplify each radical individually.
Q: Can I simplify a radical expression by using the property of radicals that states $\sqrt[n]{a^n} = a$?
A: Yes, you can simplify a radical expression by using the property of radicals that states $\sqrt[n]{a^n} = a$. For example, $\sqrt[3]{x^3} = x$.
Q: How do I simplify a radical expression with a cube root?
A: To simplify a radical expression with a cube root, you can start by factoring the numerator and denominator. Then, use the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Finally, simplify each radical individually.
Q: Can I simplify a radical expression by canceling out common factors with the cube root?
A: Yes, you can simplify a radical expression by canceling out common factors with the cube root. For example, $\sqrt[3]{\frac{4x}{5}}$ can be simplified by canceling out the common factor of $x$.
Q: How do I simplify a radical expression with a cube root and a fraction under the radical?
A: To simplify a radical expression with a cube root and a fraction under the radical, you can start by factoring the numerator and denominator. Then, use the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Finally, simplify each radical individually.
Q: Can I simplify a radical expression by using the property of radicals that states $\sqrt[n]{a^n} = a$ with the cube root?
A: Yes, you can simplify a radical expression by using the property of radicals that states $\sqrt[n]{a^n} = a$ with the cube root. For example, $\sqrt[3]{x^3} = x$.
Conclusion
Simplifying radical expressions is an important skill in mathematics, and it can be used to solve a wide range of problems. By understanding the properties of radicals and exponents, you can simplify radical expressions and solve problems with ease. Remember to always start by factoring the numerator and denominator, and then use the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Finally, simplify each radical individually to get the simplified form of the expression.
Common Mistakes to Avoid

Not factoring the numerator and denominator
Not using the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
Not simplifying each radical individually
Not canceling out common factors
Not using the property of radicals that states $\sqrt[n]{a^n} = a$

Tips and Tricks

Always start by factoring the numerator and denominator
Use the property of radicals that states $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
Simplify each radical individually
Cancel out common factors
Use the property of radicals that states $\sqrt[n]{a^n} = a$

Practice Problems

Simplify the expression $\sqrt[3]{\frac{4x}{5}}$
Simplify the expression $\sqrt[3]{\frac{20x}{5}}$
Simplify the expression $\sqrt[3]{\frac{100x}{5}}$
Simplify the expression $\sqrt[3]{\frac{100x}{125}}$

Answer Key

$\frac{2x}{\sqrt[3]{5}}$
$\frac{4x}{\sqrt[3]{5}}$
$\frac{10x}{\sqrt[3]{5}}$
$\frac{4x}{5}$