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Exponential notation is a way to express numbers using a base and an exponent. In simpler terms, an exponent tells you how many times to multiply the base by itself. For instance, in the expression \(2^3 \), the base is 2 and the exponent 3 tells you to multiply 2 by itself three times: 2 * 2 * 2 = 8.

Instead of writing out long multiplications, you can use exponential notation to save space and make calculations easier. It is especially useful when dealing with very large or very small numbers. For example, the number 1,000,000 can be written as \(10^6 \) in exponential notation.

Radicals are expressions that involve the root of a number. The most commonly known radical is the square root, but there are also cube roots, fourth roots, and so on. The general form of a radical is \(\root n \left( a \right) \), where \( n \) is the index of the root and \( a \) is the radicand. For example, \(\root 2 \left( 16 \right) \) represents the square root of 16, which is 4.

Radicals can often be converted into exponential notation, which brings us to rational exponents. For instance: \(\root n \left( a \right) \) can also be written as \( a^{1/n} \).

In our original problem, \(\root 3 \left( 18 \right) \), we had a cube root of 18. Using the rule \(\root n \left( a \right) = a^{1/n} \), we can convert this to \( 18^{1/3} \).

Understanding the rules that govern how exponents work makes it easier to manipulate and simplify expressions. Here are some crucial exponent rules:

• Product of Powers Rule: \( a^m \cdot\ a^n = a^{m+n} \) - When you multiply like bases, add the exponents.
• Power of a Power Rule: \( (a^m)^n = a^{m \cdot \ n} \) - When you raise an exponent to another exponent, multiply the exponents.
• Power of a Product Rule: \( (ab)^n = a^n \cdot\ b^n \) - When you raise a product to an exponent, distribute the exponent to each factor.
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \) - When you divide like bases, subtract the exponents.
• Zero Exponent Rule: \( a^0 = 1 \) - Any base raised to the zero power is 1.
• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \) - A negative exponent means taking the reciprocal of the base and then applying the positive exponent.