Question

select the correct answer.
what is this expression in simplified form?
$5\sqrt{2} \cdot 9\sqrt{6}$
\\(\bigcirc\\) a. $45\sqrt{2}$
\\(\bigcirc\\) b. $90\sqrt{3}$
\\(\bigcirc\\) c. $45\sqrt{3}$
\\(\bigcirc\\) d. $90$

Explanation:

Step1: Multiply coefficients and radicals separately

Multiply the coefficients \(5\) and \(9\), and multiply the radicals \(\sqrt{2}\) and \(\sqrt{6}\) using the property \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\).
\(5\sqrt{2}\cdot9\sqrt{6}=(5\times9)\cdot\sqrt{2\times6}\)

Step2: Simplify the products

Calculate \(5\times9 = 45\) and \(2\times6 = 12\), so we have \(45\sqrt{12}\).

Step3: Simplify the radical \(\sqrt{12}\)

Factor \(12\) as \(4\times3\), and use the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a\geq0,b\geq0\)). So \(\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\cdot\sqrt{3}=2\sqrt{3}\).

Step4: Multiply the coefficient with the simplified radical

Now, \(45\sqrt{12}=45\times2\sqrt{3}=90\sqrt{3}\)? Wait, no, wait. Wait, \(2\times6 = 12\), \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\), then \(45\times2\sqrt{3}\)? Wait, no, I made a mistake. Wait, \(5\times9 = 45\), \(\sqrt{2\times6}=\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\), so \(45\times2\sqrt{3}\)? No, wait, no, wait, original step: \(5\sqrt{2}\cdot9\sqrt{6}=(5\times9)\sqrt{2\times6}=45\sqrt{12}\). Then \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\), so \(45\times2\sqrt{3}=90\sqrt{3}\)? But wait, let's check again. Wait, \(2\times6 = 12\), \(\sqrt{12}=2\sqrt{3}\), so \(45\times2\sqrt{3}=90\sqrt{3}\). But wait, the options: option B is \(90\sqrt{3}\), but wait, maybe I made a mistake. Wait, no, wait, \(5\times9 = 45\), \(\sqrt{2}\times\sqrt{6}=\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\), so \(45\times2\sqrt{3}=90\sqrt{3}\). So the answer should be B? Wait, but let's check again. Wait, no, wait, \(5\sqrt{2}\times9\sqrt{6}= (5\times9)\times(\sqrt{2}\times\sqrt{6})=45\times\sqrt{12}=45\times2\sqrt{3}=90\sqrt{3}\). Yes, that's correct.

Wait, but wait, maybe I messed up. Wait, let's do it again. \(5\sqrt{2} \times 9\sqrt{6}\). Multiply the coefficients: 59=45. Multiply the radicals: sqrt(2)sqrt(6)=sqrt(12). Simplify sqrt(12): sqrt(43)=2sqrt(3). Then 452sqrt(3)=90*sqrt(3). So the correct answer is B.

Wait, but the options: A is 45√2, B is 90√3, C is 45√3, D is 90. So the correct answer is B.

Answer:

B. \( 90\sqrt{3} \)