Question

5.1 simplifying radical expressions and applying the pythagorean theorem converse, and solving right triangles pretest

simplify.
\\(\sqrt{24} + \sqrt{96}\\)

Explanation:

🆕 New Concept Discovered: Simplifying Radical Expressions
Pulling out perfect squares to combine like terms.

Step 1: Simplify the first radical

To simplify \(\sqrt{24}\), find the largest perfect square factor of \(24\), which is \(4\):

$$ \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} $$

Step 2: Simplify the second radical

To simplify \(\sqrt{96}\), find the largest perfect square factor of \(96\), which is \(16\):

$$ \sqrt{96} = \sqrt{16 \cdot 6} = \sqrt{16} \cdot \sqrt{6} = 4\sqrt{6} $$

Step 3: Combine like radicals

Now, add the simplified terms together. Since both terms share the same radical part (\(\sqrt{6}\)), you can combine them just like like terms:

$$ 2\sqrt{6} + 4\sqrt{6} = (2 + 4)\sqrt{6} = 6\sqrt{6} $$

Answer:

\(6\sqrt{6}\)