Question

which expression is equivalent to \\(\sqrt{6^2 + 3^2}\\)?
a. \\(3\sqrt{2}\\)
b. \\(9\sqrt{2}\\)
c. \\(3\sqrt{5}\\)
d. \\(9\sqrt{5}\\)
e. \\(9\\)

Explanation:

Step1: Calculate the squares

First, we calculate the values of \(6^2\) and \(3^2\).
\(6^2 = 36\) and \(3^2 = 9\).

Step2: Add the results

Next, we add these two results together: \(36 + 9 = 45\). So now our expression becomes \(\sqrt{45}\).

Step3: Simplify the square root

We can simplify \(\sqrt{45}\) by factoring 45 into its prime factors. \(45 = 9\times5\), and since \(\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}\) (by the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0,b\geq0\)), and \(\sqrt{9} = 3\), we get \(3\sqrt{5}\).

Answer:

C. \(3\sqrt{5}\)