Math Problem Statement

Sort the expressions based on which are equivalent to \(3\sqrt{2}\) or \(2\sqrt{3}\) from the options: \(\sqrt{75} - \sqrt{27}\), \(\sqrt{3} + \sqrt{3}\), \(\sqrt{6} \cdot \sqrt{3}\), \(\sqrt{50} - \sqrt{8}\).

Solution

To solve this problem, we need to simplify each expression and sort them based on which are equivalent to 323\sqrt{2} or 232\sqrt{3}.

Let's go through each expression individually:

  1. 7527\sqrt{75} - \sqrt{27}:

    • 75=25×3=53\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
    • 27=9×3=33\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}
    • So, 7527=5333=23\sqrt{75} - \sqrt{27} = 5\sqrt{3} - 3\sqrt{3} = 2\sqrt{3}

    This matches 232\sqrt{3}.

  2. 3+3\sqrt{3} + \sqrt{3}:

    • This simplifies as 13+13=231\sqrt{3} + 1\sqrt{3} = 2\sqrt{3}

    This also matches 232\sqrt{3}.

  3. 63\sqrt{6} \cdot \sqrt{3}:

    • 63=18=9×2=32\sqrt{6} \cdot \sqrt{3} = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}

    This matches 323\sqrt{2}.

  4. 508\sqrt{50} - \sqrt{8}:

    • 50=25×2=52\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
    • 8=4×2=22\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
    • So, 508=5222=32\sqrt{50} - \sqrt{8} = 5\sqrt{2} - 2\sqrt{2} = 3\sqrt{2}

    This matches 323\sqrt{2}.

Sorting Results

  • Equivalent to 323\sqrt{2}: 63\sqrt{6} \cdot \sqrt{3} and 508\sqrt{50} - \sqrt{8}
  • Equivalent to 232\sqrt{3}: 7527\sqrt{75} - \sqrt{27} and 3+3\sqrt{3} + \sqrt{3}

Would you like further details or explanations on any specific step?

Here are five related questions to explore this topic further:

  1. How do you simplify other expressions involving radicals?
  2. What rules govern the addition and subtraction of radical expressions?
  3. How can you verify if two expressions involving radicals are equivalent?
  4. What are the steps for multiplying radical expressions like ab\sqrt{a} \cdot \sqrt{b}?
  5. What are some real-world applications of simplifying radical expressions?

Tip: Remember, to add or subtract radical terms, they must have the same radicand (the number under the radical sign), similar to combining like terms in algebra.

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Math Problem Analysis

Mathematical Concepts

Radical Expressions
Simplification of Radicals
Multiplication and Addition of Radicals

Formulas

Simplification of square roots: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)
Addition of like radicals: \(a\sqrt{c} + b\sqrt{c} = (a+b)\sqrt{c}\)

Theorems

Properties of Radicals
Multiplication and Subtraction of Radical Expressions

Suitable Grade Level

Grade 9-10