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Express the following fraction in simplest form, only using positive exponents. (-5z^(-1))/(3(z^(-1))^(4)) Answer:

Express the following fraction in simplest form, only using positive exponents.\newline−5z−13(z−1)4 \frac{-5 z^{-1}}{3\left(z^{-1}\right)^{4}} \newlineAnswer:

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Full solution

Q. Express the following fraction in simplest form, only using positive exponents.\newline−5z−13(z−1)4 \frac{-5 z^{-1}}{3\left(z^{-1}\right)^{4}} \newlineAnswer:
  1. Move to Opposite Fraction: Rewrite the negative exponents as positive exponents by moving them to the opposite part of the fraction.\newline(−5z−1)(-5z^{-1}) becomes −5z-\frac{5}{z} and (z−1)4(z^{-1})^4 becomes 1z4\frac{1}{z^4} when moved to the denominator.
  2. Apply Power to Base: Apply the power to the base in the denominator.\newline(z(−1))4(z^{(-1)})^4 becomes (1/z)4(1/z)^4 which simplifies to 1/z41/z^4.
  3. Combine Expressions: Combine the expressions.\newlineNow we have (−5z)/(3z4)(-\frac{5}{z}) / (\frac{3}{z^4}). To divide fractions, we multiply by the reciprocal of the denominator.
  4. Multiply by Reciprocal: Multiply by the reciprocal of the denominator. (−5z)×(z43)(-\frac{5}{z}) \times (\frac{z^4}{3}) simplifies the expression.
  5. Cancel Common Terms: Cancel out the common zz terms.\newlineThe zz in the numerator and one zz from z4z^4 in the denominator cancel out, leaving us with −5z3-5z^3 in the numerator.
  6. Write Final Expression: Write the final simplified expression.\newlineThe final expression is −5z33.-\frac{5z^3}{3}.

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