91Ó°ÊÓ

The distributive property is an essential concept in algebra that allows us to simplify expressions. It means that we can distribute multiplication over addition or subtraction within parentheses. In the given exercise: \[(3 \sqrt{2} - \sqrt{3})(2 \sqrt{2} + 3 \sqrt{3})\]We apply the distributive property by expanding the expression, multiplying each term in the first parenthesis by each term in the second parenthesis. It looks like this: \[3 \sqrt{2} \cdot 2 \sqrt{2} + 3 \sqrt{2} \cdot 3 \sqrt{3} - \sqrt{3} \cdot 2 \sqrt{2} - \sqrt{3} \cdot 3 \sqrt{3}\]This results in four separate multiplications which we will handle in the next step. By understanding the distributive property, any complex expression can be simplified systematically.

Radical multiplication involves multiplying terms that include square roots (radicals). The fundamental rule is to multiply the numbers outside the radicals together and the numbers inside the radicals together. Here’s how it works step-by-step:

• Multiply coefficients (numbers outside the radicals).
• Multiply radicands (numbers inside the radicals).
• Simplify the resulting radicals if possible.

In our case:

• \[3 \sqrt{2} \cdot 2 \sqrt{2} = 3 \cdot 2 \cdot \sqrt{2 \cdot 2} = 6 \cdot \sqrt{4} = 6 \cdot 2 = 12\]
• \[3 \sqrt{2} \cdot 3 \sqrt{3} = 3 \cdot 3 \cdot \sqrt{2 \cdot 3} = 9 \sqrt{6}\]
• \[-\sqrt{3} \cdot 2 \sqrt{2} = -2 \sqrt{3 \cdot 2} = -2 \sqrt{6}\]
• \[-\sqrt{3} \cdot 3 \sqrt{3} = -3 \cdot 3 \cdot \sqrt{3 \cdot 3} = -9 \cdot 3 = -27\]

Using these steps, we can simplify each of the products individually. Understanding each multiplication ensures accuracy in more complex problems.

Combining like terms is a vital step in algebra to simplify expressions further. Like terms have identical variable parts. In our example, we simplify the expression by grouping and combining like terms:\[12 + 9 \sqrt{6} - 2 \sqrt{6} - 27\]Here’s the breakdown:

• Combine constants: \[12 - 27 = -15\]
• Combine like radicals: \[9 \sqrt{6} - 2 \sqrt{6} = 7 \sqrt{6}\]

After combining these, we get the final simplified expression:\[-15 + 7 \sqrt{6}\]This last step ensures that the expression is as simple as possible. By systematically combining like terms, we can easily handle even the most complex algebraic expressions.