https://file.aiquickdraw.com/mathbot/file/1728691904810hn630bdg.jpg

To rationalize the denominator and simplify the given expression, we are starting with:

\[
\frac{-3}{\sqrt[3]{5x^2}}
\]

We need to eliminate the cube root from the denominator. The cube root is \(\sqrt[3]{5x^2}\), so we multiply both the numerator and denominator by \(\sqrt[3]{5x}\), which will give us:

\[
\frac{-3 \cdot \sqrt[3]{5x}}{\sqrt[3]{5x^2} \cdot \sqrt[3]{5x}} = \frac{-3 \cdot \sqrt[3]{5x}}{\sqrt[3]{(5x^2) \cdot (5x)}} = \frac{-3 \cdot \sqrt[3]{5x}}{\sqrt[3]{5^2 x^3}}
\]

Since \(\sqrt[3]{5^2 x^3} = 5x\) (as the cube root of \(x^3\) is \(x\)), the expression simplifies to:

\[
\frac{-3 \cdot \sqrt[3]{5x}}{5x}
\]

Thus, the rationalized and simplified expression is:

\[
\frac{-3 \sqrt[3]{5x}}{5x}
\]

Would you like more details on any part of this solution?

### Related questions:
1. What are the steps to rationalize a square root denominator instead of a cube root?
2. How do you simplify cube roots in terms of variables and constants?
3. Can cube roots of negative numbers be simplified the same way?
4. How do you handle higher-order roots like fourth or fifth roots?
5. What are the common mistakes when rationalizing denominators?

### Tip: 
When rationalizing cube roots, always multiply by whatever powers of the terms in the denominator will turn them into perfect cubes. This ensures the root is eliminated from the denominator.

Rationalize the denominator and simplify: \frac{-3}{\sqrt[3]{5x^2}}, assuming all variables represent positive numbers.

Learn how to rationalize and simplify the expression \frac{-3}{\sqrt[3]{5x^2}}, including the steps to eliminate the cube root in the denominator. This detailed solution covers algebraic techniques such as rationalizing cube roots and simplifying radicals. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation