Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (z^3/4)/(z^5/4)(z^-2)

Verified step by step guidance

1. Start by rewriting the given expression clearly: \(\frac{z^{3/4}}{z^{5/4}} \cdot z^{-2}\).
2. Apply the property of exponents for division: \(\frac{a^m}{a^n} = a^{m-n}\). So, simplify \(\frac{z^{3/4}}{z^{5/4}}\) as \(z^{3/4 - 5/4} = z^{-2/4}\).
3. Simplify the exponent \(-2/4\) to \(-1/2\), so the expression becomes \(z^{-1/2} \cdot z^{-2}\).
4. Use the property of exponents for multiplication: \(a^m \cdot a^n = a^{m+n}\). Add the exponents: \(-1/2 + (-2) = -1/2 - 2 = -5/2\).
5. Rewrite the expression as \(z^{-5/2}\). Since the problem asks for no negative exponents, rewrite this as \(\frac{1}{z^{5/2}}\).

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 simplify complex expressions 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.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions by canceling common factors and applying exponent rules. When variables with exponents appear in numerator and denominator, subtracting exponents helps combine terms into a simpler form.