- To combine radicals, identify terms with identical radicands and add or subtract their coefficients.
- Simplify any radicands that contain perfect square factors, such as \(\sqrt{18} = 3\sqrt{2}\).
- For expressions where radicands remain different after simplification, the terms cannot be combined into a single radical.
- Final simplified results are: \(\boxed{15 \sqrt{7}}, \boxed{6 \sqrt{10} - 3 \sqrt{2}}, \boxed{-15 \sqrt{7} + \sqrt{61}}\).
Explanation
- Problem Analysis and Method Selection
We are presented with three expressions involving radicals and our objective is to simplify them by adding or subtracting where possible. The rule for adding or subtracting radicals is that they can be combined if and only if they share the same radicand (the number inside the square root). Specifically, \(a\sqrt{n} \pm b\sqrt{n} = (a \pm b)\sqrt{n}\). If the radicands are different, we check if they can be simplified by extracting perfect square factors. If after simplification the radicands are still different, the terms cannot be combined into a single radical.
- Simplifying the First Expression
For the first expression, \(10 \sqrt{7} + 5 \sqrt{7}\), we notice that the radicand is 7 in both terms. Since the radicands are identical, we can simply add the coefficients:
\[10 \sqrt{7} + 5 \sqrt{7} = (10 + 5) \sqrt{7} = 15 \sqrt{7}.\]
- Simplifying the Second Expression
For the second expression, \(6 \sqrt{10} - \sqrt{18}\), we first simplify \(\sqrt{18}\) because 18 contains a perfect square factor, 9:
\[\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2}.\]
Substituting this back into our expression, we get:
\[6 \sqrt{10} - 3 \sqrt{2}.\]
Since \(\sqrt{10}\) and \(\sqrt{2}\) have different radicands and \(\sqrt{10}\) (which is \(\sqrt{5 \times 2}\)) cannot be simplified to contain a simple \(\sqrt{2}\) that would allow for a single-term result without further irrational numbers, this expression is in its simplest form.
- Simplifying the Third Expression
For the third expression, \(-15 \sqrt{7} + \sqrt{61}\), we examine the radicands 7 and 61. Both are prime numbers and neither has any perfect square factors other than 1. Since the radicands are distinct and cannot be simplified further, the expression \(-15 \sqrt{7} + \sqrt{61}\) is already in its simplest form.
- Conclusion
In summary, the simplified forms of the given expressions are:
\[10 \sqrt{7} + 5 \sqrt{7} = 15 \sqrt{7}\]
\[6 \sqrt{10} - \sqrt{18} = 6 \sqrt{10} - 3 \sqrt{2}\]
\[-15 \sqrt{7} + \sqrt{61} = -15 \sqrt{7} + \sqrt{61}\]
Final results: \(\boxed{15 \sqrt{7}}, \boxed{6 \sqrt{10} - 3 \sqrt{2}}, \boxed{-15 \sqrt{7} + \sqrt{61}}\).
Examples
Understanding how to simplify radical expressions is vital in engineering and construction when dealing with dimensions that involve geometric shapes like triangles. For example, when calculating the diagonal of a rectangle or the length of a support cable using the Pythagorean theorem (\(a^2 + b^2 = c^2\)), you often end up with sums of square roots. Simplifying these allows for more manageable calculations and easier communication of measurements, ensuring structures are built to precise specifications.