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(z^(-(1)/(3)))/(z^(-(5)/(6)))=

$(z^{-(1)/(3)})/(z^{-(5)/(6)})=$

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Multiplication with rational exponents

Full solution

Q.

(z^{-(1)/(3)})/(z^{-(5)/(6)})=

Understand the problem: Understand the problem. $\newline$ We need to simplify the expression $(z^{-(1)/(3)})/(z^{-(5)/(6)})$ . This involves the division of two powers with the same base but different exponents.
Apply quotient rule: Apply the quotient rule for exponents. $\newline$ The quotient rule states that when dividing two powers with the same base, you subtract the exponents: $a^m / a^n = a^{(m-n)}$ .
Subtract exponents: Subtract the exponents. $\newline$ We have $z^{-(1)/(3)} / z^{-(5)/(6)}$ . According to the quotient rule, we subtract the exponents: $(-(1)/(3)) - (-(5)/(6))$ .
Perform subtraction: Perform the subtraction. $\newline$ Subtract the exponents: $\left(-\frac{1}{3}\right) - \left(-\frac{5}{6}\right) = \left(-\frac{1}{3}\right) + \frac{5}{6} = \frac{2}{6} - \frac{1}{3} = \frac{2}{6} - \frac{2}{6} = 0$ .
Apply to base: Apply the result of the subtraction to the base. $\newline$ Since the result of the exponent subtraction is $0$ , we have $z^0$ .
Simplify expression: Simplify the expression. $\newline$ Any number raised to the power of $0$ is $1$ . Therefore, $z^0 = 1$ .

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