91Ó°ÊÓ

Understanding exponents is fundamental in simplifying exponential expressions. An exponent, sometimes called a 'power', represents the number of times a base number is multiplied by itself. For example, in the expression \( a^n \), 'a' is the base and 'n' is the exponent. It indicates that the base 'a' is used as a factor 'n' times.

Take the number 9 in our exercise, which was raised to the power of 2, denoted as \( 9^2 \). This implies that 9 is multiplied by itself once, resulting in 81. It's a straightforward operation, but the foundation of more complex rules involving exponents. When working with these expressions, always remember to carry out exponentiation before other operations, such as multiplication or addition, unless grouping symbols like parentheses dictate otherwise.

The 'Power of a Power' rule is an important concept when dealing with nested exponents, where one exponent is raised to another. This rule states that when an exponentiated number itself is raised to an exponent, we multiply the exponents. Here's the rule: \( (a^m)^n = a^{m \times n} \).

However, in the given exercise, we have an additional step after applying the Power of a Power rule. Once we calculated the inner exponent \( 9^2 = 81 \), we do not multiply the exponents as the outer exponent \( 1/3 \) indicates taking the cube root (not cubing) of the result. Therefore, rather than multiplying the exponents directly, we recognize it as a sign to apply a different operation — the extraction of the cube root.

The cube root is a special type of root that is particularly relevant in the provided exercise. A cube root of a number x, denoted as \( \sqrt[3]{x} \) or \( x^{1/3} \), is a number that, when cubed, gives the original number x. For example, the cube root of 27 is 3 because \( 3^3 = 27 \). In the context of our example, to find \( \sqrt[3]{81} \), we are looking for a number which when multiplied by itself three times equals 81.

Upon calculating, we find that \( 81^{1/3} \approx 4.326748710 \) which cannot be simplified to an exact integer because 81 is not a perfect cube. When dealing with non-perfect cubes, it's common to round the result to a reasonable number of decimal places for practicality, generally to two decimal places as in our example, yielding 4.33.