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The cube root is a special type of root that tells us what number multiplied by itself three times will give us the original number. For example, the cube root of 27 is 3 because 3 multiplied by itself three times (3 \times 3 \times 3) equals 27. To denote a cube root, we use the radical symbol with a small 3 above it: \[\text{e.g.,} \sqrt[3]{27} = 3\].

Pulled forward from the example above, other common cube roots include:
\[ \sqrt[3]{8} = 2 \], because 2 \times 2 \times 2 = 8
\[ \sqrt[3]{64} = 4 \], because 4 \times 4 \times 4 = 64
Understanding cube roots is key for simplifying expressions with rational exponents, as we'll see next.

Simplifying exponents involves reducing the expression to its simplest form. When dealing with rational exponents, such as \((27)^{2/3}\), we need to break the problem into more manageable parts.

Step 1: Recognize that \((27)^{2/3}\) can be thought of as taking the 3rd root of 27, then squaring the result: \[\sqrt[3]{27^2}\] or \[(\sqrt[3]{27})^2\].
Step 2: Calculate the 3rd root (cube root) of 27, which is 3 (from the first section).
Step 3: Raise that result to the power of 2: \(3^{2}\).
In this case, \(3\times 3 = 9\).

This method can be used for all rational exponents, simplifying first by the denominator (the root) and then applying the numerator (the power). Always take it one step at a time for clarity and accuracy.

Rational exponents are exponents that are fractions. They actually denote both a power and a root. For instance, an exponent of \(\frac{2}{3}\) means you need to take the cube root of a number and then square it, or vice versa.

Here's the general form: \[a^{m/n} = (\sqrt[n]{a})^m\] or \[\sqrt[n]{a^m}\]

This allows us to simplify calculations that involve both roots and powers. Using our example:
Given \((27)^{2/3}\), we realize:
Step 1: Take the cube root of 27: \( \sqrt[3]{27} = 3 \).
Step 2: Square the result: \(3^2 = 9\).
This makes handling complex problems more approachable by converting between exponent and root-based solutions seamlessly.