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$Express the following fraction in simplest form, only using positive exponents. (6d^(-5)u^(-6))/((3d^(-5)u^(5))^(-1)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{6 d^{-5} u^{-6}}{\left(3 d^{-5} u^{5}\right)^{-1}}$ $\newline$ 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{6 d^{-5} u^{-6}}{\left(3 d^{-5} u^{5}\right)^{-1}}

\newline

Answer:

Invert denominator: We start by simplifying the denominator which is raised to the power of $-1$ . Raising a fraction to the power of $-1$ inverts the fraction. $\newline$ $(3d^{-5}u^{5})^{-1} = \frac{1}{3d^{-5}u^{5}}$
Multiply numerator and denominator: Now we simplify the expression by multiplying the numerator by the inverted denominator. $\newline$ $(6d^{-5}u^{-6}) \times (1/(3d^{-5}u^{5}))$
Cancel common terms: We can simplify the expression by canceling out the common terms in the numerator and the denominator. The $d^{-5}$ terms cancel each other out. $\newline$ $(6u^{-6}) \cdot \left(\frac{1}{3u^{5}}\right)$
Simplify remaining terms: Now we multiply the remaining terms. The $6$ in the numerator and the $3$ in the denominator can be simplified to $2$ (since $6/3 = 2$ ). $\newline$ $2u^{-6} \times u^{5}$
Combine $u$ terms: We combine the $u$ terms by subtracting the exponents, since they are being divided. $2u^{(-6 + 5)}$
Subtract exponents: We perform the subtraction of the exponents. $2u^{-1}$
Rewrite $u^{-1}$ : To express the fraction with only positive exponents, we rewrite $u^{-1}$ as $\frac{1}{u}$ . $\newline$ $2 \times \left(\frac{1}{u}\right)$
Express as single fraction: Finally, we write the expression as a single fraction. $\frac{2}{u}$

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