Question

select the correct answer.
what is this expression in simplified form?
$3\sqrt{3} \cdot 6\sqrt{6}$
a. $18\sqrt{3}$
b. $54\sqrt{2}$
c. $54$
d. $18\sqrt{2}$

Explanation:

Step1: Multiply the coefficients and the radicals separately

First, multiply the coefficients \(3\) and \(6\), and then multiply the radicals \(\sqrt{3}\) and \(\sqrt{6}\) using the property \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\) (for \(a\geq0,b\geq0\)).

So, \(3\sqrt{3}\cdot6\sqrt{6}=(3\times6)\cdot(\sqrt{3}\cdot\sqrt{6})\)

Calculating the product of coefficients: \(3\times6 = 18\)

Calculating the product of radicals: \(\sqrt{3}\cdot\sqrt{6}=\sqrt{3\times6}=\sqrt{18}\)

So now we have \(18\sqrt{18}\)

Step2: Simplify \(\sqrt{18}\)

We can factor \(18\) as \(9\times2\), and since \(\sqrt{9\times2}=\sqrt{9}\cdot\sqrt{2}\) (using the same property as above), and \(\sqrt{9} = 3\), so \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\)

Step3: Substitute back and simplify

Now substitute \(\sqrt{18}=3\sqrt{2}\) into \(18\sqrt{18}\):

\(18\sqrt{18}=18\times3\sqrt{2}\)

Calculating \(18\times3 = 54\), so we get \(54\sqrt{2}\)

Answer:

B. \(54\sqrt{2}\)