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Express the following fraction in simplest form, only using positive exponents. (-4(a^(-2))^(4))/(-3a^(10)) Answer:

Express the following fraction in simplest form, only using positive exponents.\newline−4(a−2)4−3a10 \frac{-4\left(a^{-2}\right)^{4}}{-3 a^{10}} \newlineAnswer:

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Full solution

Q. Express the following fraction in simplest form, only using positive exponents.\newline−4(a−2)4−3a10 \frac{-4\left(a^{-2}\right)^{4}}{-3 a^{10}} \newlineAnswer:
  1. Simplify Numerator: Simplify the numerator.\newlineThe numerator is (−4(a−2)4)(-4(a^{-2})^4). We need to apply the power to a power rule, which states that (am)n=am∗n(a^m)^n = a^{m*n}.\newline(−4(a−2)4)=−4×(a−2∗4)=−4×a−8(-4(a^{-2})^4) = -4 \times (a^{-2*4}) = -4 \times a^{-8}
  2. Simplify Denominator: Simplify the denominator.\newlineThe denominator is (−3a10)(-3a^{10}). There is nothing to simplify here, so we leave it as is.
  3. Combine Numerator and Denominator: Combine the numerator and the denominator.\newlineNow we have the fraction (−4⋅a−8)/(−3a10)(-4 \cdot a^{-8}) / (-3a^{10}). We can simplify this by dividing the coefficients and subtracting the exponents of aa using the rule am/an=a(m−n)a^m / a^n = a^{(m-n)} when m > n.\newline(−4/−3)⋅(a−8/a10)=(4/3)⋅a(−8−10)=(4/3)⋅a−18(-4/-3) \cdot (a^{-8} / a^{10}) = (4/3) \cdot a^{(-8-10)} = (4/3) \cdot a^{-18}
  4. Convert Negative Exponents: Convert negative exponents to positive exponents.\newlineWe have a−18a^{-18} in the expression, which can be written as 1/a181/a^{18} to have only positive exponents.\newline(4/3)×a−18=(4/3)×(1/a18)(4/3) \times a^{-18} = (4/3) \times (1/a^{18})
  5. Write Final Expression: Write the final simplified expression.\newlineThe final expression is (43)×(1a18)(\frac{4}{3}) \times (\frac{1}{a^{18}}), which can be written as 43a18\frac{4}{3a^{18}}.

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