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$Express the following fraction in simplest form, only using positive exponents. (4d^(-4)v^(-8))/((3d^(4))^(3)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{4 d^{-4} v^{-8}}{\left(3 d^{4}\right)^{3}}$ $\newline$ 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{4 d^{-4} v^{-8}}{\left(3 d^{4}\right)^{3}}

\newline

Answer:

Simplify Denominator: Simplify the denominator. $\newline$ The denominator is $(3d^{4})^{3}$ . When raising a power to a power, you multiply the exponents. $\newline$ $(3d^{4})^{3} = 3^{3} \times (d^{4})^{3}$ $\newline$ Calculate the powers. $\newline$ $3^{3} = 27$ and $(d^{4})^{3} = d^{(4\times3)} = d^{12}$
Rewrite with Positive Exponents: Rewrite the expression with positive exponents. $\newline$ The original expression has negative exponents in the numerator. To make them positive, we can move the terms with negative exponents from the numerator to the denominator. $\newline$ $(4d^{-4}v^{-8})/((3d^{4})^{3})$ becomes $\frac{4}{27d^{12}d^{4}v^{8}}$
Combine Like Terms: Combine like terms in the denominator. $\newline$ We have $d^{12}$ and $d^4$ in the denominator. When multiplying like bases, you add the exponents. $\newline$ $d^{12} \times d^4 = d^{(12+4)} = d^{16}$ $\newline$ Now the expression is $\frac{4}{27d^{16}v^8}$
Check for Further Simplification: Check if any further simplification is possible. The numerator is $4$ and the denominator is $27d^{16}v^{8}$ . There are no common factors between the numerator and the denominator, so no further simplification is possible.

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