Question

Simplify each expression $9 \sqrt{2(4 \sqrt{6)} }$

Answer 1

The expression to simplify is:

$9\sqrt[]{2}(4\sqrt[]{6})$

When we are multiplying two racial expressions, we multiply the constants together and the square roots together. So, the next step is:

$\begin{gathered} 9\sqrt[]{2}(4\sqrt[]{6}) \\ =(9\times4)(\sqrt[]{2}\times\sqrt[]{6}) \\ =36(\sqrt[]{2}\times\sqrt[]{6}) \end{gathered}$

Now, we an use the property

$\sqrt[]{a}\times\sqrt[]{b}=\sqrt[]{a\times b}$

to simplify it further:

$\begin{gathered} 36(\sqrt[]{2}\times\sqrt[]{6}) \\ =36(\sqrt[]{2\times6}) \\ =36\sqrt[]{12} \end{gathered}$

We can break apart the square root using the property:

$\sqrt[]{ab}=\sqrt[]{a}\sqrt[]{b}$

So, we have:

$\begin{gathered} 36\sqrt[]{12} \\ =36\sqrt[]{2}\sqrt[]{2}\sqrt[]{3} \end{gathered}$

For the final simplification, we use the property,

$\sqrt[]{a}\sqrt[]{a}=a$

The final answer is:

$\begin{gathered} 36\sqrt[]{2}\sqrt[]{2}\sqrt[]{3} \\ =36(2)\sqrt[]{3} \\ =72\sqrt[]{3} \end{gathered}$Answer$72\sqrt[]{3}$