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Express the following fraction in simplest form, only using positive exponents. (6z^(5))/(-2(z^(4))^(5)) Answer:

Express the following fraction in simplest form, only using positive exponents.\newline6z5−2(z4)5 \frac{6 z^{5}}{-2\left(z^{4}\right)^{5}} \newlineAnswer:

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Full solution

Q. Express the following fraction in simplest form, only using positive exponents.\newline6z5−2(z4)5 \frac{6 z^{5}}{-2\left(z^{4}\right)^{5}} \newlineAnswer:
  1. Simplify Denominator: Simplify the denominator using the power of a power rule.\newlineThe power of a power rule states that (am)n=am∗n(a^{m})^{n} = a^{m*n}. We apply this rule to the denominator.\newline(−2(z4)5)=−21∗5∗(z4∗5)=−25∗z20(-2(z^{4})^{5}) = -2^{1*5} * (z^{4*5}) = -2^{5} * z^{20}
  2. Calculate Value: Calculate the value of −25-2^5.\newline−25=−32-2^5 = -32
  3. Rewrite Fraction: Rewrite the original fraction with the simplified denominator.\newline(6z5)/(−2(z4)5)=(6z5)/(−32z20)(6z^{5})/(-2(z^{4})^{5}) = (6z^{5})/(-32z^{20})
  4. Divide Coefficients: Simplify the fraction by dividing the coefficients and subtracting the exponents.\newlineWhen dividing powers with the same base, subtract the exponents: zm/zn=zm−nz^{m}/z^{n} = z^{m-n}.\newline(6z5)/(−32z20)=(6/−32)⋅z5−20=−3/16⋅z−15(6z^{5})/(-32z^{20}) = (6/-32) \cdot z^{5-20} = -3/16 \cdot z^{-15}
  5. Express Negative Exponent: Express the negative exponent as a positive exponent.\newlineTo express z−15z^{-15} with a positive exponent, we write it as 1/z151/z^{15}.\newline−3/16×z−15=−3/16×1/z15-3/16 \times z^{-15} = -3/16 \times 1/z^{15}
  6. Combine Fraction: Combine the fraction.\newline−316×1z15=−316z15-\frac{3}{16} \times \frac{1}{z^{15}} = -\frac{3}{16z^{15}}

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