Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (z^1/3z^-2/3z^1/6)/(z^-1/6)³

Verified step by step guidance

1. Rewrite 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. Simplify the numerator by adding the exponents of \(z\) since the bases are the same: \(z^{\frac{1}{3} + (-\frac{2}{3}) + \frac{1}{6}}\).
3. Simplify the denominator by applying the power of a power rule: \(\left(z^{-\frac{1}{6}}\right)^3 = z^{-\frac{1}{6} \times 3} = z^{-\frac{1}{2}}\).
4. Rewrite the entire expression as a single power of \(z\) by subtracting the exponent in the denominator from the exponent in the numerator: \(z^{\left(\frac{1}{3} - \frac{2}{3} + \frac{1}{6}\right) - \left(-\frac{1}{2}\right)}\).
5. Simplify the exponent expression and rewrite the result without negative exponents by using the property \(z^{-a} = \frac{1}{z^a}\) if needed.

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 for combining and simplifying complex exponential expressions.

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, by converting them into positive exponents through reciprocals.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously, where x^(m/n) means the n-th root of x raised to the m-th power. For example, x^(1/3) is the cube root of x. Recognizing and manipulating fractional exponents is essential for simplifying expressions involving roots and powers.