Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (z^1/3z^-2/3z^1/6)/(z^-1/6)³

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\frac{z^{\frac{1}{3}} \cdot z^{-\frac{2}{3}} \cdot z^{\frac{1}{6}}}{\left(z^{-\frac{1}{6}}\right)^3}\).
2. Apply the exponent rule for powers of the same base in the numerator: add the exponents \(\frac{1}{3} + \left(-\frac{2}{3}\right) + \frac{1}{6}\) to combine the terms.
3. Simplify the denominator by applying the power of a power rule: multiply the exponents \(\left(-\frac{1}{6}\right) \times 3\) to get the new exponent for \(z\).
4. Rewrite the entire expression as a single power of \(z\) by subtracting the exponent in the denominator from the combined exponent in the numerator, using the quotient rule \(a^m / a^n = a^{m-n}\).
5. Finally, simplify the resulting exponent and rewrite the expression without any negative exponents, remembering that \(z^{-a} = \frac{1}{z^a}\) for positive \(a\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers of the same base. Key rules include adding exponents when multiplying like bases, subtracting exponents when dividing, and multiplying exponents when raising a power to another power. These rules allow combining and simplifying expressions efficiently.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-a) equals 1/x^a. Understanding this helps rewrite expressions without negative exponents, as required in the problem, by converting them into fractions with positive exponents.

Fractional Exponents and Radicals

Fractional exponents represent roots; for instance, x^(1/n) is the nth root of x. This concept allows interpreting and simplifying expressions involving roots and powers simultaneously. Recognizing fractional exponents helps in combining terms and simplifying complex expressions.