In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. 81^3/2

Verified step by step guidance

1. Identify the expression given: $81^{3/2}$.
2. Recognize that the expression is in the form of $a^{m/n}$, where $a = 81$, $m = 3$, and $n = 2$.
3. Rewrite the expression using radical notation: $\sqrt[n]{a^m}$, which becomes $\sqrt{81^3}$.
4. Simplify the expression inside the radical: $81^3$ can be rewritten as $(81^{1/2})^3$.
5. Calculate $81^{1/2}$, which is the square root of 81, and then raise the result to the power of 3.

Key Concepts

Radical Notation

Radical notation is a way to express roots of numbers using the radical symbol (√). For example, the square root of a number 'x' is written as √x. In the context of exponents, a fractional exponent indicates both a power and a root; for instance, x^(1/n) represents the nth root of x.

Exponents and Fractional Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. A fractional exponent, such as 3/2, indicates that the base should be raised to the power of 3 and then the result should be taken to the square root. This duality allows for simplification of expressions involving roots and powers.

Simplification of Expressions

Simplification involves rewriting an expression in a more manageable or concise form. This can include combining like terms, reducing fractions, or applying properties of exponents and radicals. In the case of the expression 81^(3/2), simplification would involve calculating the square root of 81 and then raising the result to the third power.