https://file.aiquickdraw.com/mathbot/file/1729133244901hk2lo5de.jpg

The expression you provided is:

\[
y \sqrt[3]{12v^3} - y \sqrt[3]{3v^2}
\]

Let's simplify this step by step.

### Step 1: Break down the cube roots.

- \(\sqrt[3]{12v^3}\) can be simplified by factoring:
  \[
  \sqrt[3]{12v^3} = \sqrt[3]{12} \cdot \sqrt[3]{v^3} = \sqrt[3]{12} \cdot v
  \]

- \(\sqrt[3]{3v^2}\) can also be simplified similarly:
  \[
  \sqrt[3]{3v^2} = \sqrt[3]{3} \cdot \sqrt[3]{v^2} = \sqrt[3]{3} \cdot v^{2/3}
  \]

### Step 2: Substitute into the original expression.

\[
y \cdot v \cdot \sqrt[3]{12} - y \cdot v^{2/3} \cdot \sqrt[3]{3}
\]

### Step 3: Factor out the common term \(y\).

\[
y \left( v \cdot \sqrt[3]{12} - v^{2/3} \cdot \sqrt[3]{3} \right)
\]

At this stage, the expression has been simplified as much as possible given the cube roots and the powers of \(v\). The simplified expression is:

\[
y \left( v \cdot \sqrt[3]{12} - v^{2/3} \cdot \sqrt[3]{3} \right)
\]

Let me know if you'd like further clarifications!

### Questions for further exploration:
1. How would the expression change if the powers of \(v\) were different?
2. What happens if you replace the cube roots with square roots?
3. Can you combine the cube roots further if given specific values for \(v\)?
4. What if \(y\) were not part of the equation? How would that affect the simplification?
5. How does the simplification change if \(v\) is a specific number?

### Tip:
When simplifying expressions with cube roots, always factor both the constants and the variables to their respective powers, and remember that cube roots simplify powers of three effectively.

Simplify as much as possible: y * ∛(12v^3) - y * ∛(3v^2). Assume that all variables represent positive real numbers.

Learn how to simplify the expression y * ∛(12v^3) - y * ∛(3v^2) step-by-step, including factorization and properties of cube roots. This algebra problem is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation