Question

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

Explanation:

Step1: Multiply the coefficients and the radicals separately

First, multiply the coefficients (the numbers outside the square roots): \(5 \times 9 = 45\). Then, multiply the radicals (the square root parts): \(\sqrt{2} \times \sqrt{6}\). Using the property of square roots \(\sqrt{a} \times \sqrt{b}=\sqrt{ab}\), we get \(\sqrt{2\times6}=\sqrt{12}\). So now we have \(45\sqrt{12}\).

Step2: Simplify the radical \(\sqrt{12}\)

We can factor \(12\) into \(4\times3\), and since \(\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}\) (using the same square root property as before), and \(\sqrt{4} = 2\), this simplifies to \(2\sqrt{3}\). Now substitute this back into our expression: \(45\times2\sqrt{3}\).

Step3: Multiply the remaining coefficients

Multiply \(45\) and \(2\): \(45\times2 = 90\). So the simplified form is \(90\sqrt{3}\).

Answer:

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