Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (z^3/4)/(z^5/4)(z^-2)

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Start by writing the expression clearly: \(\frac{z^{3/4}}{z^{5/4}} \cdot z^{-2}\).

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 \(z^{3/4 - 5/4} = z^{-2/4}\).

Simplify the exponent \(-2/4\) to \(-1/2\), so the expression becomes \(z^{-1/2} \cdot z^{-2}\).

Use the property of exponents for multiplication: \(a^m \cdot a^n = a^{m+n}\). Add the exponents: \(-1/2 + (-2) = -1/2 - 2\).

Combine the exponents to get \(z^{-5/2}\). Finally, rewrite the expression without negative exponents by using the rule \(a^{-m} = \frac{1}{a^m}\), so the simplified expression is \(\frac{1}{z^{5/2}}\).

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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 allow simplification of 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 helps rewrite expressions without negative exponents, as required in the problem.

Simplifying Expressions with Fractional Exponents

Fractional exponents represent roots and powers simultaneously, such as x^(m/n) meaning the n-th root of x raised to the m-th power. Simplifying expressions with fractional exponents involves applying exponent rules carefully while interpreting roots and powers correctly.

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (z3/4)/(z5/4)(z-2)