Question

simplify.
\sqrt{45} \times 2\sqrt{80}

Explanation:

Step1: Simplify each square root

First, we simplify \(\sqrt{45}\) and \(\sqrt{80}\) by factoring out perfect squares.
For \(\sqrt{45}\), we have \(45 = 9\times5\), so \(\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}\) (since \(\sqrt{9} = 3\)).
For \(\sqrt{80}\), we have \(80 = 16\times5\), so \(\sqrt{80}=\sqrt{16\times5}=\sqrt{16}\times\sqrt{5}=4\sqrt{5}\) (since \(\sqrt{16}=4\)).

Step2: Substitute the simplified roots back into the expression

The original expression is \(\sqrt{45}\times2\sqrt{80}\). Substituting the simplified forms, we get:
\(3\sqrt{5}\times2\times4\sqrt{5}\)

Step3: Multiply the coefficients and the square roots separately

First, multiply the coefficients: \(3\times2\times4 = 24\).
Then, multiply the square roots: \(\sqrt{5}\times\sqrt{5}=\sqrt{5\times5}=\sqrt{25} = 5\) (by the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)).

Step4: Multiply the results from Step3

Now, multiply the coefficient result and the square root result: \(24\times5 = 120\).

Answer:

\(120\)