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$Express the following fraction in simplest form using only positive exponents. (12s^(6))/((4s^(4))^(3)) Answer:$

Express the following fraction in simplest form using only positive exponents. $\newline$ $\frac{12 s^{6}}{\left(4 s^{4}\right)^{3}}$ $\newline$ 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{12 s^{6}}{\left(4 s^{4}\right)^{3}}

\newline

Answer:

Write Expression: Write down the original expression. $\newline$ We have the fraction $(\frac{12s^{6}}{(4s^{4})^{3}})$ .
Simplify Denominator: Simplify the denominator. $\newline$ The denominator is $(4s^{4})^{3}$ . When raising a power to a power, you multiply the exponents. So, $(4s^{4})^{3}$ becomes $4^{3} \times (s^{4})^{3}$ .
Calculate Denominator Powers: Calculate the powers in the denominator. $\newline$ $4^{3}$ is $4 \times 4 \times 4$ , which equals $64$ . $(s^{4})^{3}$ is $s^{4\times3}$ , which equals $s^{12}$ . $\newline$ So, the denominator becomes $64s^{12}$ .
Rewrite with Simplified Denominator: Rewrite the fraction with the simplified denominator. $\newline$ Now we have $(12s^{6})/(64s^{12})$ .
Simplify Fraction: Simplify the fraction by dividing the coefficients and subtracting the exponents. $\newline$ Divide $12$ by $64$ to get $\frac{3}{16}$ (since $12$ is $3$ times $4$ and $64$ is $16$ times $4$ ). Subtract the exponents in $s^{6}$ and $64$ $0$ to get $64$ $1$ , which is $64$ $2$ . $\newline$ So, the fraction simplifies to $64$ $3$ .
Express Negative Exponent: Express the negative exponent as a positive exponent. $\newline$ To express $s^{-6}$ with a positive exponent, we write it as $1/s^{6}$ . So, the fraction becomes $(3/16)\cdot(1/s^{6})$ .
Combine Fraction: Combine the fraction. $\newline$ The final simplified form of the fraction is $\frac{3}{16s^{6}}$ .

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