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Understanding the concept of the cube root is key when solving expressions with fractional exponents like the one in this exercise. A cube root of a number is a value that, when multiplied by itself three times, returns the original number. Think of it as "undoing" the cube. For example, the cube root of 8 is 2, because if you multiply 2 × 2 × 2, you get 8 again.

To denote a cube root mathematically, we use the radical symbol and write it as \(\sqrt[3]{x}\). In simpler terms, finding the cube root is the reverse operation of cubing a number. This helps when dealing with exponents where the base number is cubed to reach a specific value.

Recognizing cube roots can make calculations easier and faster, especially for small numbers and powers you memorize. For example:

• The cube root of 1 is 1
• The cube root of 8 is 2
• The cube root of 27 is 3

Utilizing cube roots is vital in further mathematics and solidifying your understanding of numbers and their powers.

Exponentiation is a mathematical operation where a number (the base) is multiplied by itself a certain number of times, as indicated by the exponent. It's like repeated multiplication. When we see \(8^{2/3}\) in the exercise, it involves both a cube root and a squaring operation.

The base in this expression is 8, while the exponent is \(\frac{2}{3}\). To handle this type of exponent, you consider breaking it into two parts: first, apply the cube root (from the \(\frac{1}{3}\) part of the exponent), then square the result (from the \(2\) part of the exponent).

Here's a basic rundown of exponentiation:

• \(2^3\) indicates multiplying 2 by itself, three times, which equals 8
• \(4^2\) means multiplying 4 by itself, which results in 16
• For fractions like \(8^{2/3}\), perform operations step by step for easier comprehension

Exponentiation can help solve complex problems efficiently when you follow the rules designed to simplify calculations, like utilizing the order of operations (PEMDAS/BODMAS).