Full solution

Q. Simplify. Express your answer using positive exponents.

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\frac{4z}{z^8}

Write Given Expression: Write down the given expression. $\newline$ The given expression is $\frac{4z}{z^8}$ . We need to simplify this expression by dividing the terms.
Apply Quotient Rule: Apply the quotient rule for exponents. $\newline$ The quotient rule states that when dividing like bases, you subtract the exponents. In this case, we have $z$ to the power of $1$ in the numerator and $z$ to the power of $8$ in the denominator. $\newline$ $\frac{4z}{z^8} = 4 \times z^{(1-8)}$
Subtract Exponents: Subtract the exponents. $\newline$ Now we subtract the exponents of $z$ . $\newline$ $z^{1-8} = z^{-7}$
Simplify Expression: Simplify the expression. $\newline$ We multiply the coefficient $4$ by the result of the exponent subtraction. $\newline$ $4 \times z^{(-7)} = 4z^{(-7)}$
Express Positive Exponent: Express the negative exponent as a positive exponent. $\newline$ Since the question asks for the answer to be expressed using positive exponents, we need to rewrite the negative exponent as a positive exponent. This is done by taking the reciprocal of the base with the negative exponent. $\newline$ $4z^{-7} = \frac{4}{z^7}$