In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible.5xz^-⅓

Verified step by step guidance

1. Identify the expression with a negative exponent: \( z^{-\frac{1}{3}} \).
2. Rewrite the expression with a positive exponent by using the property \( a^{-n} = \frac{1}{a^n} \).
3. Apply this property to \( z^{-\frac{1}{3}} \) to get \( \frac{1}{z^{\frac{1}{3}}} \).
4. Combine the rewritten expression with the other terms: \( 5x \cdot \frac{1}{z^{\frac{1}{3}}} \).
5. Express the entire expression as a single fraction: \( \frac{5x}{z^{\frac{1}{3}}} \).

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, x^-n can be rewritten as 1/x^n. This concept is crucial for transforming expressions with negative exponents into a more manageable form, allowing for simplification and easier manipulation in algebraic operations.

Rational Exponents

Rational exponents express roots in exponential form. An exponent of the form 1/n indicates the nth root of a number. For instance, x^(1/3) represents the cube root of x. Understanding rational exponents is essential for rewriting expressions and simplifying them, especially when dealing with roots and powers.

Simplification of Algebraic Expressions

Simplification involves rewriting an expression in a more concise or manageable form, often by combining like terms or reducing fractions. In the context of exponents, this may include applying the laws of exponents to combine terms or eliminate negative exponents. Mastery of simplification techniques is vital for solving algebraic problems efficiently.