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Rationalizing the denominator is a key technique in simplifying radical expressions. Typically, when you have a fraction with a radical in the denominator, the process of "rationalizing" involves eliminating the radical from the denominator by multiplying both the numerator and the denominator by a suitable form of one. This transformation results in an equivalent expression that does not have any radicals in the denominator.

For instance, if the denominator was \( \sqrt{b} \), you would multiply both the top and bottom of the fraction by \( \sqrt{b} \). This transforms the denominator into \( b \), effectively removing the radical. In our example exercise, however, the given expression \( 2 \sqrt[5]{2} \) has no denominator requiring rationalization, meaning it is already in a simplified form.

Key points to remember about rationalizing denominators include:

• Aim to have whole numbers in the denominator when expressed in simplest form.
• Multiply by the appropriate form of one to keep the expression equivalent.
• Simplified radicals should have no fractions under the radical.

Simplifying radicals is the process of expressing a radical in its simplest form. Whether it's a square root, cube root, or any nth root, the objective is to break it down to its most basic form.

To simplify a radical like \( \sqrt[5]{64} \), you need to find a perfect power that matches the index. In this case, since \( 64 = 2^6 \), and we're working with a fifth root, you simplify it to \( 2 \times \sqrt[5]{2} \). The expression is now devoid of unnecessary complexity, thus in its simplest form.

Some useful strategies for simplifying radicals include:

• Factor the number or expression inside the radical until a perfect power that matches the root index emerges.
• Use prime factorization when unsure about the factors of a given number.
• Remember that the goal is to reduce the radicals to the smallest number possible inside.

Multiplying radicals involves combining the expressions under a single radical when they have the same root index. This process is handy because it can significantly simplify complex-looking expressions.

For example, in our provided exercise, the roots \( \sqrt[5]{4} \) and \( \sqrt[5]{16} \) can be multiplied into \( \sqrt[5]{4 \times 16} \). This multiplication results in \( \sqrt[5]{64} \). From here, simplifying gives \( 2 \times \sqrt[5]{2} \).

When multiplying radicals, always keep these points in mind:

• Ensure the radicals have the same index before attempting to combine them.
• Always simplify the resulting expression to its lowest terms.
• Recheck your work after simplifying to confirm it can no longer be reduced.