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

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{\left(2 w^{-4}\right)^{3}}{-3 w^{-3}}$ $\newline$ 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{\left(2 w^{-4}\right)^{3}}{-3 w^{-3}}

\newline

Answer:

Rewrite with positive exponents: Rewrite the expression with positive exponents. $\newline$ We have the expression $\frac{(2w^{-4})^3}{-3w^{-3}}$ . To simplify, we first rewrite the negative exponents as positive by moving the terms to the opposite side of the fraction. $\newline$ $(2^3 \cdot w^{-4\cdot3}) / (-3 \cdot w^{-3})$ $\newline$ $= (8 \cdot w^{-12}) / (-3 \cdot w^{-3})$ $\newline$ $= (\frac{8}{-3}) \cdot (\frac{w^{-12}}{w^{-3}})$
Simplify numerical part: Simplify the numerical part of the fraction. $\newline$ We simplify the numerical part of the fraction by dividing $8$ by $-3$ . $\newline$ $(8 / -3) = -8/3$
Apply quotient rule for exponents: Apply the quotient rule for exponents. $\newline$ We simplify the variable part of the fraction using the quotient rule for exponents, which states that $a^{m}/a^{n} = a^{(m-n)}$ when m > n. $\newline$ $w^{(-12)} / w^{(-3)} = w^{(-12 - (-3))} = w^{(-12 + 3)} = w^{(-9)}$
Rewrite with positive exponents: Rewrite the expression with positive exponents. $\newline$ We have $w^{-9}$ , which we can rewrite with a positive exponent by moving it to the denominator. $\newline$ $w^{-9} = \frac{1}{w^9}$
Combine numerical and variable parts: Combine the numerical and variable parts. $\newline$ Now we combine the numerical part from Step $2$ and the variable part from Step $4$ . $\newline$ $-\frac{8}{3} \times \frac{1}{w^9} = -\frac{8}{3w^9}$

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