Simplify.
$\sqrt{45}-5 \sqrt{20}$
The given expression to simplify is \(\sqrt{45} - 5\sqrt{20}\). To simplify this, we need to extract any perfect square factors from the numbers under the square roots. A perfect square factor is a number like 4, 9, 16, 25, and so on. Extracting these allows us to use the property \(\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}\) to simplify the radicals into a common form, which will then allow us to combine like terms.
First, examine the term \(\sqrt{45}\). We can factor 45 as \(9 \times 5\). Since 9 is a perfect square, we can write:
\[\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}.\]
This simplifies the first part of the expression.
Next, examine the term \(\sqrt{20}\). We can factor 20 as \(4 \times 5\). Since 4 is a perfect square, we can write:
\[\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}.\]
Substituting this back into the original expression gives:
\[\sqrt{45} - 5\sqrt{20} = 3\sqrt{5} - 5(2\sqrt{5}).\]
Finally, perform the multiplication and subtraction:
\[3\sqrt{5} - 5(2\sqrt{5}) = 3\sqrt{5} - 10\sqrt{5}.\]
Combining the like terms, we subtract the coefficients:
\[(3 - 10)\sqrt{5} = -7\sqrt{5}.\]
Thus, the expression simplifies to \(\boxed{-7\sqrt{5}}\).
Understanding how to simplify radical expressions is essential in geometry, such as calculating the perimeter or distance between points. For example, if you are designing a park with paths forming a right triangle, the lengths of the paths might involve square roots. Simplifying these values helps in accurately measuring materials needed for construction. By reducing complex radicals to their simplest form, you make calculations manageable and precise.