Write each root using exponents and evaluate. ∜81

Key Concepts

Roots and Exponents

Roots and exponents are fundamental concepts in algebra that describe the relationship between numbers. The nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the square root of 9 is 3 because 3^2 = 9. This relationship can also be expressed using fractional exponents, where the nth root of a number 'a' is represented as a^(1/n).

Evaluating Roots

Evaluating roots involves finding the value that satisfies the root equation. For instance, to evaluate the fourth root of 81, we seek a number that, when raised to the fourth power, equals 81. This process may involve recognizing perfect powers or using prime factorization to simplify the evaluation. In this case, since 81 = 3^4, the fourth root is 3.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing complex expressions to their simplest form. This can include factoring, combining like terms, and applying properties of exponents and roots. For example, when dealing with roots, one can express them in terms of exponents to facilitate easier calculations and comparisons, such as rewriting ∜81 as 81^(1/4).