Verified Answer :

  • Identify perfect square factors for each radicand: \(\sqrt{45} = 3\sqrt{5}\) and \(\sqrt{80} = 4\sqrt{5}\).
  • Substitute these back into the original expression: \(-3(3\sqrt{5}) + 4\sqrt{5}\).
  • Simplify the product to \(-9\sqrt{5} + 4\sqrt{5}\).
  • Combine like terms to obtain \(\boxed{-5\sqrt{5}}\).

Explanation

  1. Analyzing the Expression

We are asked to simplify the expression \(-3\sqrt{45} + \sqrt{80}\). The strategy is to simplify each radical term individually by identifying the largest perfect square factor within the radicands (the numbers inside the square root signs), then combine the resulting terms.

  1. Simplifying the Radical Terms

First, look at \(\sqrt{45}\). The factors of 45 include 1, 3, 5, 9, 15, 45. The largest perfect square factor is 9. We can rewrite the radical as:
\[\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}.\]
Next, look at \(\sqrt{80}\). The factors of 80 include 1, 2, 4, 5, 8, 10, 16, 20, 40, 80. The largest perfect square factor is 16. We can rewrite the radical as:
\[\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}.\]

  1. Combining Terms and Final Result

Now, substitute these simplified forms back into the original expression \(-3\sqrt{45} + \sqrt{80}\):
\[-3(3\sqrt{5}) + 4\sqrt{5}.\]
Performing the multiplication in the first term:
\[-9\sqrt{5} + 4\sqrt{5}.\]
Finally, combine the like terms:
\[(-9 + 4)\sqrt{5} = -5\sqrt{5}.\]
The simplified result is \(-5\sqrt{5}\).

Examples

Simplifying radicals is a fundamental skill in geometry and physics. For example, when calculating the diagonal length of a rectangle with side lengths of \(\sqrt{45}\) and \(\sqrt{80}\) using the Pythagorean theorem, you would need to calculate \(\sqrt{(\sqrt{45})^2 + (\sqrt{80})^2} = \sqrt{45 + 80} = \sqrt{125} = 5\sqrt{5}\). Being able to manipulate these expressions efficiently helps in solving for dimensions and distances in real-world engineering or construction scenarios.