Perform the indicated operations. Assume all variables represent positive real numbers. (3√2 + √3) (2√3 - √2)

Verified step by step guidance

1. Identify the expression to be simplified: \( (3\sqrt{2} + \sqrt{3})(2\sqrt{3} - \sqrt{2}) \). This is a product of two binomials involving square roots.
2. Apply the distributive property (FOIL method) to multiply each term in the first binomial by each term in the second binomial: \( (3\sqrt{2})(2\sqrt{3}) + (3\sqrt{2})(-\sqrt{2}) + (\sqrt{3})(2\sqrt{3}) + (\sqrt{3})(-\sqrt{2}) \)
3. Simplify each product by multiplying the coefficients and the square roots separately, remembering that \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \): - \( 3 \times 2 = 6 \) and \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \) - \( 3 \times (-1) = -3 \) and \( \sqrt{2} \times \sqrt{2} = \sqrt{4} \) - \( 1 \times 2 = 2 \) and \( \sqrt{3} \times \sqrt{3} = \sqrt{9} \) - \( 1 \times (-1) = -1 \) and \( \sqrt{3} \times \sqrt{2} = \sqrt{6} \)
4. Rewrite the expression with the simplified terms: \( 6\sqrt{6} - 3\sqrt{4} + 2\sqrt{9} - \sqrt{6} \)
5. Simplify the square roots of perfect squares (\( \sqrt{4} = 2 \), \( \sqrt{9} = 3 \)), then combine like terms (terms with \( \sqrt{6} \)) to write the expression in its simplest form.

Key Concepts

Operations with Radicals

Radicals represent roots, such as square roots, and can be manipulated using arithmetic operations. When multiplying expressions with radicals, apply distributive property and simplify by combining like terms and using properties of roots.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is essential for multiplying binomials, including those with radicals, by multiplying each term in the first expression by each term in the second.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the root to its simplest form and combining like terms. For example, √a * √b = √(ab), and perfect squares under the root can be simplified to integers.