Question - Simplifying a Radical Expression

The given expression is \( \sqrt[3]{-9} \times \sqrt[3]{\sqrt[3]{3}} \).

Let's simplify the expression step by step.

\( \sqrt[3]{-9} \) can be written as \( -\sqrt[3]{9} \) because \( \sqrt[3]{-1} \) equals -1.

Now, \( -\sqrt[3]{9} \times \sqrt[3]{\sqrt[3]{3}} \) can be simplified further.

Since \( \sqrt[3]{9} = \sqrt[3]{3^2} \), we can replace \( \sqrt[3]{9} \) with \( 3^{2/3} \), giving us:

\( -3^{2/3} \times \sqrt[3]{\sqrt[3]{3}} \)

We now focus on \( \sqrt[3]{\sqrt[3]{3}} \), which simplifies to \( \sqrt[3]{3^{1/3}} \).

Using the property \( \sqrt[n]{a^m} = a^{m/n} \), we get:

\( \sqrt[3]{3^{1/3}} = 3^{(1/3)(1/3)} = 3^{1/9} \).

So the expression becomes:

\( -3^{2/3} \times 3^{1/9} \)

Now, we combine the exponents of the same base (3) using the law of exponents \( a^m \times a^n = a^{m+n} \):

\( -3^{2/3 + 1/9} \)

We find a common denominator for the exponents:

\( -3^{6/9 + 1/9} \)

\( -3^{7/9} \)

Therefore, the simplified expression is \( -3^{7/9} \).