Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. -81^3/4

Verified step by step guidance

1. Identify the expression: $-{81}^{3/4}$.
2. Recognize that the expression involves a fractional exponent, which can be rewritten using radicals: $-{81}^{3/4}={(-81)}^{3/4}$.
3. Rewrite the expression using the property of exponents: $a^{m/n}=\sqrt[n]{a^m}$.
4. Apply the property: $-{81}^{3/4}={\left(\sqrt[4]{-81}\right)}^3$.
5. Simplify $\sqrt[4]{-81}$ and then raise the result to the power of 3.

Key Concepts

Exponents and Radicals

Exponents represent repeated multiplication of a base number. For example, a number raised to a fractional exponent, such as 3/4, indicates both a root and a power: the denominator (4) signifies the root, while the numerator (3) indicates the power. Understanding how to manipulate these expressions is crucial for simplification.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For instance, x^(-n) equals 1/(x^n). In the context of this problem, simplifying expressions without negative exponents means rewriting any terms with negative exponents as positive fractions.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, eliminating negative exponents, and applying the laws of exponents. This process is essential for clarity and ease of further calculations, especially when dealing with complex expressions.