Multiply.
$\sqrt{2}(3 \sqrt{10}-\sqrt{3})$
Simplify your answer as much as possible.
The expression to simplify is \(\sqrt{2}(3\sqrt{10} - \sqrt{3})\). To solve this, we use the distributive property of multiplication over subtraction, which states that \(a(b - c) = ab - ac\). Applying this property, we can write:
\[\sqrt{2}(3\sqrt{10} - \sqrt{3}) = \sqrt{2} \cdot 3\sqrt{10} - \sqrt{2} \cdot \sqrt{3}\]
Next, we simplify each term individually using the property of radicals \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) for \(a, b \geq 0\).
For the first term, we have:
\[3(\sqrt{2} \cdot \sqrt{10}) = 3\sqrt{2 \cdot 10} = 3\sqrt{20}\]
For the second term, we have:
\[\sqrt{2 \cdot 3} = \sqrt{6}\]
Combining these, the expression becomes \(3\sqrt{20} - \sqrt{6}\).
Finally, we simplify \(\sqrt{20}\) by factoring out the largest perfect square:
\[\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}\]
Substituting this back into our expression:
\[3(2\sqrt{5}) - \sqrt{6} = 6\sqrt{5} - \sqrt{6}\]
Since \(\sqrt{5}\) and \(\sqrt{6}\) cannot be simplified further, this is the final simplified form.
The process of distributing and simplifying radical expressions is very common in geometry, particularly when calculating the lengths of sides in triangles or diagonals of rectangles using the Pythagorean theorem. For instance, if you were determining the dimensions of a garden plot or the length of a support beam, you might encounter expressions involving square roots. Simplifying these expressions allows for much easier comparisons and measurements, ensuring precision in construction and design projects.