What Is The Simplified Form Of The Following Expression? Assume $x \neq 0$.A. $\sqrt[5]{\frac{10 X}{3 X^3}}$B. $ 120 X 3 5 3 X \frac{\sqrt[5]{120 X^3}}{3 X} 3 X 5 120 X 3 [/tex]C. $\frac{\sqrt[5]{10 X}}{3 X}$D.

Mar 11, 2025 by ADMIN 221 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

Understanding the Problem

When dealing with radical expressions, it's essential to simplify them to their most basic form. This involves applying the rules of exponents and radicals to reduce the expression to a more manageable form. In this article, we'll focus on simplifying the given expression: $\sqrt[5]{\frac{10 x}{3 x^3}}$. We'll explore the different options and determine which one is the simplified form of the expression.

Option A: $\sqrt[5]{\frac{10 x}{3 x^3}}$

To simplify this expression, we need to apply the rule of radicals, which states that $\sqrt[n]a^n} = a$. We can rewrite the expression as $\sqrt[5]{\frac{10 x3 x^3}} = \frac{\sqrt[5]{10 x}}{\sqrt[5]{3 x^3}}$. Now, we can simplify the numerator and denominator separately. The numerator can be rewritten as $\sqrt[5]{10 x = \sqrt[5]10} \cdot \sqrt[5]{x}$. The denominator can be rewritten as $\sqrt[5]{3 x^3 = \sqrt[5]3} \cdot \sqrt[5]{x^3} = \sqrt[5]{3} \cdot x \cdot \sqrt[5]{x^2}$. Now, we can simplify the expression by canceling out the common factors $\frac{\sqrt[5]{10 x}\sqrt[5]{3 x^3}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{x}}{\sqrt[5]{3} \cdot x \cdot \sqrt[5]{x^2}} = \frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x \cdot \sqrt[5]{x^2}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10}\sqrt[5]{3} \cdot x \cdot \sqrt[5]{x^2}} = \frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x \cdot x \cdot \sqrt[5]{x}} = \frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x^2 \cdot \sqrt[5]{x}}$. Now, we can simplify the expression by canceling out the common factors $\frac{\sqrt[5]{10}\sqrt[5]{3} \cdot x^2 \cdot \sqrt[5]{x}} = \frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x^2 \cdot x^{2/5}} = \frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x^{7/5}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10}\sqrt[5]{3} \cdot x^{7/5}} = \frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x^{7/5}} \cdot \frac{x^{-7/5}}{x{-7/5}} = \frac{\sqrt[5]{10} \cdot x^{-7/5}}{\sqrt[5]{3} \cdot x^{7/5} \cdot x^{-7/5}} = \frac{\sqrt[5]{10} \cdot x^{-7/5}}{\sqrt[5]{3}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10 \cdot x^-7/5}}{\sqrt[5]{3}} = \frac{\sqrt[5]{10} \cdot x^{-7/5}}{\sqrt[5]{3}} \cdot \frac{\sqrt[5]{3}}{\sqrt[5]{3}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3}}{\sqrt[5]{3} \cdot \sqrt[5]{3} \cdot x^{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3}}{\sqrt[5]{3^2} \cdot x^{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3}}{\sqrt[5]{9} \cdot x^{7/5}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10 \cdot \sqrt[5]3}}{\sqrt[5]{9} \cdot x^{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3}}{\sqrt[5]{9} \cdot x^{7/5}} \cdot \frac{\sqrt[5]{9}}{\sqrt[5]{9}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9}}{\sqrt[5]{9} \cdot \sqrt[5]{9} \cdot x^{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9}}{\sqrt[5]{9^2} \cdot x^{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9}}{\sqrt[5]{81} \cdot x^{7/5}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10 \cdot \sqrt[5]3} \cdot \sqrt[5]{9}}{\sqrt[5]{81} \cdot x^{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9}}{\sqrt[5]{81} \cdot x^{7/5}} \cdot \frac{x^{7/5}}{x{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot x^{7/5}}{\sqrt[5]{81} \cdot x^{7/5} \cdot x^{7/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot x^{7/5}}{\sqrt[5]{81} \cdot x^{14/5}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10 \cdot \sqrt[5]3} \cdot \sqrt[5]{9} \cdot x^{7/5}}{\sqrt[5]{81} \cdot x^{14/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot x^{7/5}}{\sqrt[5]{81} \cdot x^{14/5}} \cdot \frac{\sqrt[5]{81}}{\sqrt[5]{81}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot \sqrt[5]{81} \cdot x^{{7/5}}{\sqrt[5]{81}2} \cdot x^{14/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot \sqrt[5]{81} \cdot x^{7/5}}{\sqrt[5]{6561} \cdot x^{14/5}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10 \cdot \sqrt[5]3} \cdot \sqrt[5]{9} \cdot \sqrt[5]{81} \cdot x^{7/5}}{\sqrt[5]{6561} \cdot x^{14/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot \sqrt[5]{81} \cdot x^{7/5}}{\sqrt[5]{6561} \cdot x^{14/5}} \cdot \frac{x^{-14/5}}{x{-14/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot \sqrt[5]{81} \cdot x^{7/5} \cdot x^{-14/5}}{\sqrt[5]{6561} \cdot x^{14/5} \cdot x^{-14/5}} = \frac{\sqrt[5]{10} \cdot \sqrt[5]{3} \cdot \sqrt[5]{9} \cdot \sqrt[5]{81} \cdot x^{-7/5}}{\sqrt[5]{6561}}$. However, we can simplify this expression further by canceling out the common factors $\frac{\sqrt[5]{10 \cdot \sqrt[5]{3}

Understanding the Problem

Option A: $\sqrt[5]{\frac{10 x}{3 x^3}}$

Q&A: Simplifying Radical Expressions

Q: What is the simplified form of the expression $\sqrt[5]{\frac{10 x}{3 x^3}}$?

A: The simplified form of the expression is $\frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x \cdot \sqrt[5]{x^2}}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to apply the rule of radicals, which states that $\sqrt[n]a^n} = a$. You can rewrite the expression as $\sqrt[n]{a = \sqrt[n]{a} \cdot \frac{\sqrt[n]{a}}{\sqrt[n]{a}} = \frac{\sqrt[n]{a^2}}{\sqrt[n]{a}}$.

Q: What is the difference between a radical and an exponent?

A: A radical and an exponent are both used to represent repeated multiplication, but they are used in different ways. A radical is used to represent repeated multiplication of a number, while an exponent is used to represent repeated multiplication of a number by itself.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you need to apply the rule of radicals, which states that $\sqrt[n]a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$. You can rewrite the expression as $\sqrt[n]{a \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$.

Q: What is the simplified form of the expression $\frac{\sqrt[5]{10 x}}{\sqrt[5]{3 x^3}}$?

A: The simplified form of the expression is $\frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x \cdot \sqrt[5]{x^2}}$.

Q: How do I simplify an expression with a radical in the denominator?

A: To simplify an expression with a radical in the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator is the same as the denominator, but with the opposite sign.

Q: What is the simplified form of the expression $\frac{\sqrt[5]{10 x}}{\sqrt[5]{3 x^3}}$?

A: The simplified form of the expression is $\frac{\sqrt[5]{10}}{\sqrt[5]{3} \cdot x \cdot \sqrt[5]{x^2}}$.

Conclusion

Simplifying radical expressions is an essential skill in mathematics. By applying the rules of exponents and radicals, you can simplify complex expressions and make them more manageable. In this article, we've explored the different options and determined which one is the simplified form of the expression. We've also answered some common questions about simplifying radical expressions.