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$Express the following fraction in simplest form, only using positive exponents. (-12k^(-10))/(3(k^(-5))^(4)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{-12 k^{-10}}{3\left(k^{-5}\right)^{4}}$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

Q. Express the following fraction in simplest form, only using positive exponents.

\newline

\frac{-12 k^{-10}}{3\left(k^{-5}\right)^{4}}

\newline

Answer:

Simplify Denominator: Simplify the denominator. $\newline$ We have a power raised to a power in the denominator, which means we multiply the exponents. $\newline$ $(3(k^{-5})^{4}) = 3 \times (k^{-5 \times 4}) = 3 \times k^{-20}$
Rewrite with Positive Exponents: Rewrite the fraction with positive exponents. $\newline$ To convert negative exponents to positive, we take the reciprocal of the base raised to the positive exponent. $\newline$ $(-12k^{-10})/(3k^{-20}) = (-12/k^{10})/(3/k^{20})$
Divide and Subtract Exponents: Simplify the fraction by dividing the coefficients and subtracting the exponents. $\newline$ When dividing powers with the same base, we subtract the exponents. $\newline$ $(-12/3) \times (k^{20-10}) = -4 \times k^{10}$
Write Final Answer: Write the final answer. $\newline$ The fraction is now simplified to $-4k^{10}$ with positive exponents.

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