In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible.16^-¾

Verified step by step guidance

1. Identify the base and the exponent in the expression: The base is 16 and the exponent is -\frac{3}{4}.
2. Rewrite the expression with a positive exponent by taking the reciprocal of the base: \left(\frac{1}{16}\right)^{\frac{3}{4}}.
3. Recognize that raising a number to the \frac{3}{4} power is equivalent to taking the fourth root and then cubing the result.
4. Express the fourth root of 16: \sqrt[4]{16}.
5. Cube the result of the fourth root: \left(\sqrt[4]{16}\right)^3.

Key Concepts

Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal. For example, a term like a^-n can be rewritten as 1/a^n. This concept is essential for transforming expressions with negative exponents into a more manageable form, particularly when simplifying or rewriting them with positive exponents.

Rational Exponents

Rational exponents express roots and powers in a unified way. An exponent in the form of m/n indicates the n-th root of the base raised to the m-th power. For instance, x^(1/2) represents the square root of x. Understanding this concept allows for the conversion between radical expressions and exponent forms.

Simplification of Exponents

Simplifying expressions with exponents involves applying the laws of exponents, such as the product, quotient, and power rules. This process can include combining like terms, reducing fractions, and rewriting expressions in their simplest form. Mastery of these rules is crucial for effectively manipulating and simplifying algebraic expressions.