Perform the indicated operations. Assume all variables represent positive real numbers. $3\sqrt{72m^2} - 5\sqrt{32m^2} - 3\sqrt{18m^2}$

Verified step by step guidance

1. Start by expressing each radical term in the form of \( \sqrt[3]{a \cdot b} \), where \(a\) is a perfect cube and \(b\) is the remaining factor. For example, rewrite \( \sqrt[3]{72m^2} \) as \( \sqrt[3]{\text{(perfect cube)} \times \text{(other factor)}} \).
2. Identify the perfect cube factors inside each cube root. For instance, since \(72 = 8 \times 9\) and \(8\) is a perfect cube (\(2^3\)), rewrite \( \sqrt[3]{72m^2} \) as \( \sqrt[3]{8 \times 9m^2} \). Do the same for the other terms: \( \sqrt[3]{32m^2} \) and \( \sqrt[3]{18m^2} \).
3. Use the property of cube roots that \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \) to separate the perfect cube and the remaining factor. For example, \( \sqrt[3]{8 \times 9m^2} = \sqrt[3]{8} \times \sqrt[3]{9m^2} \).
4. Simplify the cube roots of the perfect cubes. Since \( \sqrt[3]{8} = 2 \), \( \sqrt[3]{27} = 3 \), and so on, replace these with their simplified values. This will allow you to rewrite each term as a product of a constant and a cube root of the remaining factor.
5. After simplifying each term, combine like terms by subtracting or adding the coefficients of the cube roots that have the same radicand (the expression inside the cube root). This will give you the simplified expression for the original problem.

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing the radicand as a product of perfect squares and other factors, then taking the square root of the perfect squares outside the radical. This process makes it easier to combine like terms and perform operations on radicals.

Like Radicals and Combining Terms

Like radicals have the same radicand and index, allowing their coefficients to be added or subtracted directly. Recognizing and rewriting radicals to have the same radicand is essential for combining terms in expressions involving roots.

Operations with Variables under Radicals

When variables are under radicals, their exponents affect simplification. For example, √(m²) simplifies to m if m is positive. Understanding how to handle variables inside radicals helps in correctly simplifying and combining radical expressions.