Simplify cube root of fifth root of 6

Solution:

Step 1:

Express $\sqrt[3]{\sqrt[5]{6}}$ as a single radical by multiplying the indices, resulting in $\sqrt[15]{6}$.

Step 2:

This expression can be represented in various ways. In its exact form, it remains $\sqrt[15]{6}$. To express it as a decimal, it approximates to $1.12687761 \ldots$

Knowledge Notes:

To simplify nested radicals, such as a cube root of a fifth root, you can use the property of exponents that states $(a^{m})^{n} = a^{m \cdot n}$. In the context of radicals, this means that the cube root of the fifth root of a number is the same as the fifteenth root of that number, since $3 \times 5 = 15$.

The cube root of a number $a$ is written as $\sqrt[3]{a}$, and the fifth root of a number $b$ is written as $\sqrt[5]{b}$. If you have a cube root of a fifth root, you can combine them by multiplying the root indices (3 and 5) to get the fifteenth root of the number, which is written as $\sqrt[15]{a}$ if $a$ is the number inside the fifth root.

The exact form of a radical expression is the one that contains the radical sign and the number inside it, without any decimal or fractional approximation. In this case, the exact form is $\sqrt[15]{6}$.

The decimal form of a radical expression is a numerical approximation of the radical's value. Since most radicals do not result in a perfect integer or a simple fraction, a decimal approximation is often used for practical calculations. This approximation can be found using a calculator or a computer algebra system.

In summary, the key knowledge points involved in this solution are:

• Properties of exponents and radicals
• Simplifying nested radicals by multiplying indices
• Representing radical expressions in exact and decimal forms