Perform the indicated operations. Assume all variables represent positive real numbers. 3√72m² - 5√32m² - 3√18m²

Verified step by step guidance

1. Step 1: Simplify each square root term separately. Start with \( \sqrt{72m^2} \).
2. Step 2: Break down \( \sqrt{72m^2} \) into \( \sqrt{72} \times \sqrt{m^2} \). Since \( \sqrt{m^2} = m \), focus on simplifying \( \sqrt{72} \).
3. Step 3: Factor 72 into its prime factors: \( 72 = 2^3 \times 3^2 \). Therefore, \( \sqrt{72} = \sqrt{2^3 \times 3^2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} \). Thus, \( \sqrt{72m^2} = 6m\sqrt{2} \).
4. Step 4: Repeat the process for \( \sqrt{32m^2} \) and \( \sqrt{18m^2} \). For \( \sqrt{32m^2} \), factor 32 as \( 2^5 \), so \( \sqrt{32} = 4\sqrt{2} \), giving \( \sqrt{32m^2} = 4m\sqrt{2} \). For \( \sqrt{18m^2} \), factor 18 as \( 2 \times 3^2 \), so \( \sqrt{18} = 3\sqrt{2} \), giving \( \sqrt{18m^2} = 3m\sqrt{2} \).
5. Step 5: Substitute the simplified terms back into the expression: \( 3(6m\sqrt{2}) - 5(4m\sqrt{2}) - 3(3m\sqrt{2}) \). Combine like terms by factoring out \( m\sqrt{2} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are essential in simplifying expressions. Understanding how to manipulate these expressions, including combining like terms and simplifying radicals, is crucial for solving problems that involve them.

Simplifying Radicals

Simplifying radicals means rewriting a radical expression in its simplest form. This often involves factoring out perfect squares or cubes from under the radical sign. For example, √72 can be simplified to 6√2, which makes calculations easier and clearer.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves adding or subtracting terms that have the same variable and exponent. In the context of radical expressions, this means only combining terms that have the same radical part, which is necessary for simplifying the overall expression.