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$Express the following fraction in simplest form using only positive exponents. ((5u^(4))^(3))/(2u^(4)) Answer:$

Express the following fraction in simplest form using only positive exponents. $\newline$ $\frac{\left(5 u^{4}\right)^{3}}{2 u^{4}}$ $\newline$ 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{\left(5 u^{4}\right)^{3}}{2 u^{4}}

\newline

Answer:

Apply Power Rule: Apply the power of a power rule to the numerator. $\newline$ The power of a power rule states that $(a^m)^n = a^{m*n}$ . We apply this rule to the numerator $(5u^4)^3$ . $\newline$ $(5u^4)^3 = 5^3 \times (u^4)^3$ $\newline$ = $125$ \times u^{ $4$ * $3$ } $\newline$ = $125$ \times u^{ $12$ }
Rewrite Fraction: Rewrite the fraction with the simplified numerator. $\newline$ Now we have the fraction as: $\newline$ $(125 \cdot u^{12}) / (2u^4)$
Apply Quotient Rule: Apply the quotient of powers rule to the fraction. $\newline$ The quotient of powers rule states that $a^m / a^n = a^{(m-n)}$ when $a \neq 0$ . We apply this rule to the $u$ terms in the fraction. $\newline$ $(125 * u^{12}) / (2u^4) = (125/2) * (u^{12-4})$ $\newline$ $= (125/2) * u^8$
Simplify Constant: Simplify the constant term if possible. $\newline$ The constant term $\frac{125}{2}$ cannot be simplified further as it is already in its simplest form. $\newline$ So the final simplified fraction is: $\newline$ $\left(\frac{125}{2}\right) \cdot u^8$

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