Simplify the expression. Write your answer in simplest form.
$8\sqrt{5}+\sqrt{10}+\sqrt{5}=$
The expression to simplify is \(8\sqrt{5} + \sqrt{10} + \sqrt{5}\). We observe that there are terms involving \(\sqrt{5}\) and a term involving \(\sqrt{10}\). Like radicals are those that have the same radicand; in this case, \(8\sqrt{5}\) and \(\sqrt{5}\) are like terms, while \(\sqrt{10}\) is a distinct radical term. To simplify, we should combine the like radicals.
We use the distributive property, which states that \(a\sqrt{x} + b\sqrt{x} = (a+b)\sqrt{x}\). Applying this to the terms involving \(\sqrt{5}\), we have:
\[8\sqrt{5} + 1\sqrt{5} = (8+1)\sqrt{5} = 9\sqrt{5}.\]
The constant coefficient of the second \(\sqrt{5}\) term is implied to be \(1\).
Now we substitute this back into the original expression:
\[(8\sqrt{5} + \sqrt{5}) + \sqrt{10} = 9\sqrt{5} + \sqrt{10}.\]
Since \(\sqrt{10} = \sqrt{2 \cdot 5} = \sqrt{2}\sqrt{5}\), one could write the expression as \(\sqrt{5}(9 + \sqrt{2})\), but the standard simplified form for these types of sums is keeping the radicals separated where they cannot be combined further. Thus, the expression is in its simplest form.
Simplifying radical expressions is a fundamental skill in algebra that appears when calculating side lengths using the Pythagorean theorem, dealing with complex engineering formulas, or even in computer science algorithms involving geometry. For example, if you calculate the perimeter of a complex shape where segments are given as multiples of \(\sqrt{5}\) and \(\sqrt{10}\), grouping those values correctly allows you to estimate and report measurements efficiently. Simplifying expressions like \(9\sqrt{5} + \sqrt{10}\) is essential for maintaining precision throughout long calculations.