Math Problem Statement

Fully simplify 3√75 + √45 + 2√20.

Solution

To simplify the expression 375+45+2203\sqrt{75} + \sqrt{45} + 2\sqrt{20}, let's follow these steps:

  1. Simplify each square root by factoring out perfect squares.
  • For 75\sqrt{75}: 75=25×3=53\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} So, 375=3×53=1533\sqrt{75} = 3 \times 5\sqrt{3} = 15\sqrt{3}.

  • For 45\sqrt{45}: 45=9×5=35\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}

  • For 2202\sqrt{20}: 20=4×5=25\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} So, 220=2×25=452\sqrt{20} = 2 \times 2\sqrt{5} = 4\sqrt{5}.

  1. Combine like terms: 153+35+4515\sqrt{3} + 3\sqrt{5} + 4\sqrt{5} Combine 353\sqrt{5} and 454\sqrt{5}: 153+7515\sqrt{3} + 7\sqrt{5}

Thus, the simplified expression is: 153+7515\sqrt{3} + 7\sqrt{5}

Would you like further clarification or have any questions about this simplification?

Here are some related questions:

  1. How do we simplify square roots with perfect square factors?
  2. What is the difference between like and unlike terms in radical expressions?
  3. How do we add or subtract square roots with different radicands?
  4. How do we simplify higher-order square roots?
  5. What is the method for rationalizing the denominator when working with square roots?

Tip: Always check for perfect square factors first when simplifying square roots. It makes the process much quicker!

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Math Problem Analysis

Mathematical Concepts

Simplification of square roots
Combination of like terms
Factoring perfect squares

Formulas

√(a × b) = √a × √b
Simplify: a√n + b√n = (a + b)√n

Theorems

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Suitable Grade Level

Grades 8-10