Question

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

Explanation:

Step1: Multiply the coefficients and the radicals separately

First, multiply the coefficients (the numbers outside the square roots) and then multiply the radicals (the square root parts) separately. So, \(3\sqrt{3} \cdot 6\sqrt{6}=(3\times6)\times(\sqrt{3}\times\sqrt{6})\)

Step2: Calculate the product of coefficients and simplify the radical product

The product of the coefficients \(3\times6 = 18\). For the radicals, use the property \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\), so \(\sqrt{3}\times\sqrt{6}=\sqrt{3\times6}=\sqrt{18}\). Then, simplify \(\sqrt{18}\). We know that \(18 = 9\times2\), so \(\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}\) (since \(\sqrt{9} = 3\)).

Step3: Multiply the results from Step1 and Step2

Now, we have \(18\times3\sqrt{2}\) (because from Step2, \(\sqrt{18}=3\sqrt{2}\) and the coefficient product was 18). Calculate \(18\times3 = 54\), so the expression simplifies to \(54\sqrt{2}\)

Answer:

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