https://file.aiquickdraw.com/mathbot/file/1727750477841th45h57l.jpg

The question asks to express \( \frac{2x^2}{\sqrt{3x^7}} \) in its simplest radical form with a rational denominator.

Let's break it down step by step:

### Step 1: Simplify the denominator
The denominator is \( \sqrt{3x^7} \). We can rewrite it as:
\[
\sqrt{3x^7} = \sqrt{3} \cdot \sqrt{x^7} = \sqrt{3} \cdot x^{7/2}
\]

### Step 2: Simplify the entire expression
Now, the expression becomes:
\[
\frac{2x^2}{\sqrt{3} \cdot x^{7/2}}
\]
We can simplify the powers of \(x\):
\[
\frac{x^2}{x^{7/2}} = x^{2 - 7/2} = x^{-3/2}
\]
So the expression becomes:
\[
\frac{2x^{-3/2}}{\sqrt{3}} = \frac{2}{\sqrt{3}} \cdot x^{-3/2}
\]

### Step 3: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{3} \):
\[
\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\]
Thus, the simplified form is:
\[
\frac{2\sqrt{3}}{3} \cdot x^{-3/2}
\]

This is the simplest radical form.

### Final Answer:
\[
\frac{2\sqrt{3}}{3}x^{-3/2}
\]

Would you like further details on any step or have any additional questions? Here are some related questions to explore:

1. How do you simplify radical expressions with variables?
2. What are the steps to rationalize a denominator?
3. How do you handle fractional exponents during simplification?
4. Can this process be applied to more complex radicals?
5. How does rationalizing a denominator affect the value of the expression?

**Tip:** Always check for opportunities to combine like terms when simplifying expressions with exponents!

Express 2x^2 / sqrt(3x^7) in simplest radical form with a rational denominator (assume x > 0).

Learn how to simplify the expression 2x^2 / sqrt(3x^7) into its simplest radical form with a rational denominator. This step-by-step guide covers key concepts such as rationalizing the denominator, handling exponents, and simplifying radicals. Suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation