Verified Answer :

  • Simplify each radical: \(\sqrt{72} = 6\sqrt{2}\), \(\sqrt{75} = 5\sqrt{3}\), and \(\sqrt{18} = 3\sqrt{2}\).
  • Substitute these back: \(5(6\sqrt{2}) + 3(5\sqrt{3}) - 4(3\sqrt{2})\).
  • Distribute coefficients: \(30\sqrt{2} + 15\sqrt{3} - 12\sqrt{2}\).
  • Combine like terms to get \(\boxed{18\sqrt{2} + 15\sqrt{3}}\).

Explanation

  1. Understanding the Problem and Data Analysis

We are given the radical expression \(5 \sqrt{72} + 3 \sqrt{75} - 4 \sqrt{18}\). Our objective is to simplify this expression by simplifying each radical term individually and then combining like terms. A radical \(\sqrt{n}\) is simplified by finding the largest perfect square that divides \(n\), say \(k^2\), and writing it as \(\sqrt{k^2 \cdot m} = k\sqrt{m}\).

  1. Simplifying Each Radical Term

First, we simplify each radical term:

  • For \(\sqrt{72}\): Since \(72 = 36 \times 2\) and 36 is a perfect square, we have \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\).
  • For \(\sqrt{75}\): Since \(75 = 25 \times 3\) and 25 is a perfect square, we have \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\).
  • For \(\sqrt{18}\): Since \(18 = 9 \times 2\) and 9 is a perfect square, we have \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).
  1. Combining Terms and Finding the Simplified Form

Now, we substitute these simplified forms back into the original expression:
\[5(6\sqrt{2}) + 3(5\sqrt{3}) - 4(3\sqrt{2})\]
Perform the multiplications:
\[30\sqrt{2} + 15\sqrt{3} - 12\sqrt{2}\]
Next, we combine the like terms, which are the terms involving \(\sqrt{2}\):
\[(30 - 12)\sqrt{2} + 15\sqrt{3} = 18\sqrt{2} + 15\sqrt{3}\]
The expression cannot be simplified further as \(\sqrt{2}\) and \(\sqrt{3}\) are not like radicals.

Examples

Simplifying radical expressions is a foundational skill in algebra, useful in geometry when dealing with distances and areas. For example, if you need to calculate the exact total length of two structural supports in a bridge design that are measured in radical form (e.g., \(\sqrt{72}\) and \(\sqrt{18}\) meters), simplifying them allows you to see if the lengths can be combined or if they represent different scaling factors, ensuring precision in engineering and construction projects.

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