Simplify.
$\sqrt{50}+5 \sqrt{98}$
We are asked to simplify the expression \(\sqrt{50} + 5\sqrt{98}\). To simplify radical expressions of this form, the strategy is to identify the largest perfect square factor within each number under the square root. Once the perfect squares are extracted, we will have like terms involving \(\sqrt{2}\), which can then be combined.
First, consider the term \(\sqrt{50}\). We can factor 50 into a perfect square and another factor:
\[50 = 25 \times 2 = 5^2 \times 2\]
Using the property of radicals \(\sqrt{a^2b} = a\sqrt{b}\), we can rewrite this as:
\[\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\]
Now, consider the second term, \(5\sqrt{98}\). We can factor 98 as:
\[98 = 49 \times 2 = 7^2 \times 2\]
So, the radical part is:
\[\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\]
Substituting this back into the original expression:
\[5\sqrt{98} = 5 \times (7\sqrt{2}) = 35\sqrt{2}\]
Now we combine the simplified terms:
\[\sqrt{50} + 5\sqrt{98} = 5\sqrt{2} + 35\sqrt{2}\]
Since both terms contain \(\sqrt{2}\), we can add them by summing their coefficients:
\[(5 + 35)\sqrt{2} = 40\sqrt{2}\]
The expression is now in its simplest form.
Simplifying radicals is a fundamental skill in geometry and physics. For example, if you need to calculate the diagonal of a rectangle with sides \(\sqrt{50}\) and \(5\sqrt{98}\) units, you would often encounter expressions that need simplification before you can compare them or use them in further calculations. Mastering these simplifications helps in handling complex formulas efficiently and accurately.