91Ó°ÊÓ

The cube root of a number is a value that, when multiplied by itself three times, equals the original number. This special root is denoted by the radical symbol with a small three above it, as in \(\text{ }\text{ }\text{ }\text{ }\sqrt[3]{x}\) for a number \(x\). For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).

In our exercise, we're dealing with the cube root of 12, which is a bit trickier since 12 is not a perfect cube. Nevertheless, grasping the concept of the cube root is essential for understanding rational exponents and simplifying expressions that involve them.

To improve comprehension, visualize a cube whose volume is the number in question. The cube root gives you the length of one side of this imaginary cube. It's helpful to know that every positive number has exactly one real cube root, which can also be expressed using rational exponents.

The nth root extends the concept of square and cube roots. It refers to a number that must be multiplied by itself \(n\) times to yield the original number. Represented as \(\text{ }\text{ }\text{ }\text{ }\sqrt[n]{x}\), where \(n\) is the degree of the root and \(x\) is the radicand, the nth root can be any integer—although when \(n\) is 2, it is more commonly referred to as a square root.

It's important to remember that if \(n\) is even and \(x\) is positive, there will be two real nth roots: one positive and one negative. However, if \(x\) is negative, there are no real nth roots. When \(n\) is odd, there's exactly one real nth root regardless of the sign of \(x\).

The exercise involves converting this concept into rational exponent form. Doing so provides an alternative way of expressing roots that can make certain calculations and algebraic manipulations easier.

Exponentiation rules, or laws of exponents, are vital in manipulating expressions involving powers and roots. They allow us to perform operations like multiplication and division of terms with exponents, raising a power to another power, or converting roots to rational exponents.

Here are some key exponentiation rules that relate to our exercise:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(a^m \text{ } / \text{ } a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^n = a^n \times b^n\)
• Power of a Quotient: \((a/b)^n = a^n \text{ } / \text{ } b^n\)
• Negative Exponent: \(a^{-n} = 1 \text{ } / \text{ } a^n\)

Understanding these rules is essential because they are frequently applied in algebra to simplify expressions and solve equations. For instance, the cube root of 12 expressed as a rational exponent (\(12^{1/3}\)) relies on the exponentiation rule that connects roots and rational exponents: an nth root of a can be written as \(a^{1/n}\).