Question

question
what is the product of \\(\sqrt{6}\\) and \\(9\sqrt{12}\\) in simplest radical form?
answer
\\(\square\\)
submit answer
\\(\sqrt{}\\)

Explanation:

Step1: Multiply the coefficients and radicands

We have \(\sqrt{6} \times 9\sqrt{12}\). First, multiply the coefficients (the number outside the square root) and the radicands (the number inside the square root) separately. The coefficient of \(\sqrt{6}\) is \(1\) and the coefficient of \(9\sqrt{12}\) is \(9\), so multiplying the coefficients gives \(1\times9 = 9\). Multiplying the radicands gives \(\sqrt{6\times12}\). So now we have \(9\sqrt{6\times12}\).

Step2: Simplify the radicand

Calculate \(6\times12 = 72\), so we have \(9\sqrt{72}\). Now we need to simplify \(\sqrt{72}\). We can factor \(72\) into a perfect square and another number. \(72 = 36\times2\), and \(\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}\) (by the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0,b\geq0\)). Since \(\sqrt{36} = 6\), then \(\sqrt{72}=6\sqrt{2}\).

Step3: Multiply the coefficient with the simplified radical

Now we have \(9\times6\sqrt{2}\). Multiplying \(9\) and \(6\) gives \(54\). So the product in simplest radical form is \(54\sqrt{2}\).

Answer:

\(54\sqrt{2}\)