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$Express the following fraction in simplest form using only positive exponents. (2b^(5))/(2(b^(5))^(4)) Answer:$

Express the following fraction in simplest form using only positive exponents. $\newline$ $\frac{2 b^{5}}{2\left(b^{5}\right)^{4}}$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

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

\newline

\frac{2 b^{5}}{2\left(b^{5}\right)^{4}}

\newline

Answer:

Simplify Exponents: Simplify the expression $(2b^{5})/(2(b^{5})^{4})$ by first looking at the exponents. $\newline$ We have a power to a power situation in the denominator, which means we multiply the exponents. $\newline$ $(b^{5})^{4} = b^{5*4} = b^{20}$
Rewrite Fraction: Rewrite the fraction with the simplified exponent in the denominator. $\newline$ $(2b^{5})/(2(b^{5})^{4})$ becomes $(2b^{5})/(2b^{20})$
Cancel Common Factors: Cancel out the common factors in the numerator and the denominator. $\newline$ The number $2$ in the numerator and denominator cancels out, and we can use the quotient rule for exponents to simplify $b^{5}/b^{20}$ . $\newline$ $(2b^{5})/(2b^{20})$ becomes $b^{5-20}$ after canceling the $2$ s.
Calculate New Exponent: Calculate the new exponent for $b$ after subtracting the exponents. $b^{(5-20)} = b^{(-15)}$
Express Negative Exponent: Since we want only positive exponents, we need to express $b^{-15}$ as a positive exponent. $\newline$ $b^{-15}$ is the same as $1/(b^{15})$ because a negative exponent indicates the reciprocal.
Write Final Expression: Write the final simplified expression. $\newline$ The fraction $(2b^{5})/(2(b^{5})^{4})$ simplifies to $1/(b^{15})$ .

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