Radical Simplification of $\sqrt{54} + \sqrt{150}$

First, simplify the known expression $\sqrt{54} + \sqrt{150}$. Find the largest perfect square factor for each number under the square root.

• $\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}$
• $\sqrt{150} = \sqrt{25 \times 6} = \sqrt{25} \times \sqrt{6} = 5\sqrt{6}$

Add the simplified terms:

$\sqrt{54} + \sqrt{150} = 3\sqrt{6} + 5\sqrt{6} = 8\sqrt{6}$

We are given that $8\sqrt{6} \approx 19.60$.

Target Radical Sum: $\sqrt{216} + \sqrt{96}$

Next, simplify the expression $\sqrt{216} + \sqrt{96}$ in a similar way.

• $\sqrt{216} = \sqrt{36 \times 6} = \sqrt{36} \times \sqrt{6} = 6\sqrt{6}$
• $\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}$

Add these simplified terms:

$\sqrt{216} + \sqrt{96} = 6\sqrt{6} + 4\sqrt{6} = 10\sqrt{6}$

Calculating the Value

We need to find the value of $10\sqrt{6}$. We know that $8\sqrt{6} \approx 19.60$. We can set up a ratio or scale the known value.

Since $10\sqrt{6} = \frac{10}{8} \times (8\sqrt{6})$, substitute the known approximate value:

$10\sqrt{6} \approx \frac{10}{8} \times 19.60$

$10\sqrt{6} \approx 1.25 \times 19.60$

$10\sqrt{6} \approx 24.5$

The value is $24.5$, which is already correct to one decimal place.