Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (k^1/3)/(k^2/3)(k^-1)

Verified step by step guidance

1. Start by writing the expression clearly: \(\frac{k^{\frac{1}{3}}}{k^{\frac{2}{3}}} \cdot k^{-1}\).
2. Use the property of exponents for division: \(\frac{a^m}{a^n} = a^{m-n}\). Apply this to the fraction part: \(k^{\frac{1}{3} - \frac{2}{3}}\).
3. Simplify the exponent in the numerator: \(\frac{1}{3} - \frac{2}{3} = -\frac{1}{3}\), so the expression becomes \(k^{-\frac{1}{3}} \cdot k^{-1}\).
4. Use the property of exponents for multiplication: \(a^m \cdot a^n = a^{m+n}\). Add the exponents: \(-\frac{1}{3} + (-1) = -\frac{1}{3} - 1\).
5. Simplify the sum of exponents and rewrite the expression with a positive exponent by using \(a^{-m} = \frac{1}{a^m}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include multiplying powers with the same base by adding exponents, dividing powers by subtracting exponents, and raising a power to another power by multiplying exponents. These rules allow simplification of expressions like the given problem.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-n) = 1/x^n. Understanding this helps rewrite expressions without negative exponents, as required in the problem.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For instance, x^(1/3) means the cube root of x. Recognizing fractional exponents helps in combining and simplifying terms with rational powers.