Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (k^1/3)/(k^2/3)(k^-1)

Verified step by step guidance

1. Start with the given expression: \(\frac{k^{\frac{1}{3}}}{k^{\frac{2}{3}}} \cdot k^{-1}\).
2. Apply the property of exponents for division: \(\frac{a^m}{a^n} = a^{m-n}\). So, simplify \(\frac{k^{\frac{1}{3}}}{k^{\frac{2}{3}}}\) as \(k^{\frac{1}{3} - \frac{2}{3}}\).
3. Calculate the exponent subtraction: \(\frac{1}{3} - \frac{2}{3} = -\frac{1}{3}\), 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 as \(k^{-(\frac{1}{3} + 1)} = k^{-\frac{4}{3}}\). Finally, rewrite without negative exponents by expressing it as \(\frac{1}{k^{\frac{4}{3}}}\).

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 by subtracting exponents, and raising a power to another power by multiplying exponents. These rules help combine and simplify terms efficiently.

Negative Exponents

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

Fractional Exponents

Fractional exponents represent roots and powers simultaneously; for instance, x^(m/n) means the nth root of x raised to the mth power. Recognizing and manipulating fractional exponents is essential for simplifying expressions involving roots and powers.